Properties

Degree 8
Conductor $ 3^{4} $
Sign $1$
Motivic weight 16
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.05e3·3-s + 1.24e5·4-s − 3.14e6·7-s + 1.14e7·9-s − 2.56e8·12-s − 1.58e9·13-s + 3.94e9·16-s − 5.61e10·19-s + 6.44e9·21-s + 3.01e11·25-s − 1.26e11·27-s − 3.92e11·28-s + 2.47e12·31-s + 1.42e12·36-s + 3.70e11·37-s + 3.24e12·39-s − 2.80e13·43-s − 8.10e12·48-s − 4.67e13·49-s − 1.97e14·52-s + 1.15e14·57-s + 3.62e14·61-s − 3.58e13·63-s − 4.23e14·64-s − 2.67e13·67-s + 3.17e14·73-s − 6.17e14·75-s + ⋯
L(s)  = 1  − 0.312·3-s + 1.90·4-s − 0.544·7-s + 0.265·9-s − 0.595·12-s − 1.93·13-s + 0.919·16-s − 3.30·19-s + 0.170·21-s + 1.97·25-s − 0.448·27-s − 1.03·28-s + 2.89·31-s + 0.505·36-s + 0.105·37-s + 0.606·39-s − 2.40·43-s − 0.287·48-s − 1.40·49-s − 3.69·52-s + 1.03·57-s + 1.88·61-s − 0.144·63-s − 1.50·64-s − 0.0659·67-s + 0.393·73-s − 0.617·75-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(17-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+8)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(81\)    =    \(3^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(16\)
character  :  induced by $\chi_{3} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 81,\ (\ :8, 8, 8, 8),\ 1)$
$L(\frac{17}{2})$  $\approx$  $1.83003$
$L(\frac12)$  $\approx$  $1.83003$
$L(9)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 3$, \(F_p\) is a polynomial of degree 8. If $p = 3$, then $F_p$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p$
bad3$D_{4}$ \( 1 + 76 p^{3} T - 122 p^{10} T^{2} + 76 p^{19} T^{3} + p^{32} T^{4} \)
good2$C_2^2 \wr C_2$ \( 1 - 15605 p^{3} T^{2} + 11364297 p^{10} T^{4} - 15605 p^{35} T^{6} + p^{64} T^{8} \)
5$C_2^2 \wr C_2$ \( 1 - 12045550276 p^{2} T^{2} + 20208485552534687982 p^{5} T^{4} - 12045550276 p^{34} T^{6} + p^{64} T^{8} \)
7$D_{4}$ \( ( 1 + 224396 p T + 79000079514 p^{3} T^{2} + 224396 p^{17} T^{3} + p^{32} T^{4} )^{2} \)
11$C_2^2 \wr C_2$ \( 1 - 5004710841163244 p T^{2} + \)\(15\!\cdots\!26\)\( p^{3} T^{4} - 5004710841163244 p^{33} T^{6} + p^{64} T^{8} \)
13$D_{4}$ \( ( 1 + 60797324 p T + 8193950692930518 p^{2} T^{2} + 60797324 p^{17} T^{3} + p^{32} T^{4} )^{2} \)
17$C_2^2 \wr C_2$ \( 1 - \)\(13\!\cdots\!60\)\( T^{2} + \)\(81\!\cdots\!58\)\( T^{4} - \)\(13\!\cdots\!60\)\( p^{32} T^{6} + p^{64} T^{8} \)
19$D_{4}$ \( ( 1 + 1476766220 p T + \)\(42\!\cdots\!98\)\( T^{2} + 1476766220 p^{17} T^{3} + p^{32} T^{4} )^{2} \)
23$C_2^2 \wr C_2$ \( 1 - \)\(90\!\cdots\!80\)\( p T^{2} + \)\(18\!\cdots\!78\)\( T^{4} - \)\(90\!\cdots\!80\)\( p^{33} T^{6} + p^{64} T^{8} \)
29$C_2^2 \wr C_2$ \( 1 - \)\(48\!\cdots\!44\)\( T^{2} + \)\(17\!\cdots\!66\)\( T^{4} - \)\(48\!\cdots\!44\)\( p^{32} T^{6} + p^{64} T^{8} \)
31$D_{4}$ \( ( 1 - 39867438004 p T + \)\(18\!\cdots\!06\)\( T^{2} - 39867438004 p^{17} T^{3} + p^{32} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 - 185281606948 T + \)\(24\!\cdots\!22\)\( T^{2} - 185281606948 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
41$C_2^2 \wr C_2$ \( 1 - \)\(23\!\cdots\!04\)\( T^{2} + \)\(21\!\cdots\!66\)\( T^{4} - \)\(23\!\cdots\!04\)\( p^{32} T^{6} + p^{64} T^{8} \)
43$D_{4}$ \( ( 1 + 14032511031332 T + \)\(22\!\cdots\!02\)\( T^{2} + 14032511031332 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
47$C_2^2 \wr C_2$ \( 1 - \)\(18\!\cdots\!60\)\( T^{2} + \)\(14\!\cdots\!98\)\( T^{4} - \)\(18\!\cdots\!60\)\( p^{32} T^{6} + p^{64} T^{8} \)
53$C_2^2 \wr C_2$ \( 1 - \)\(12\!\cdots\!60\)\( T^{2} + \)\(71\!\cdots\!98\)\( T^{4} - \)\(12\!\cdots\!60\)\( p^{32} T^{6} + p^{64} T^{8} \)
59$C_2^2 \wr C_2$ \( 1 - \)\(16\!\cdots\!04\)\( T^{2} + \)\(93\!\cdots\!66\)\( T^{4} - \)\(16\!\cdots\!04\)\( p^{32} T^{6} + p^{64} T^{8} \)
61$D_{4}$ \( ( 1 - 181134896541604 T + \)\(81\!\cdots\!26\)\( T^{2} - 181134896541604 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 + 13387202681732 T + \)\(30\!\cdots\!42\)\( T^{2} + 13387202681732 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
71$C_2^2 \wr C_2$ \( 1 - \)\(12\!\cdots\!44\)\( T^{2} + \)\(66\!\cdots\!66\)\( T^{4} - \)\(12\!\cdots\!44\)\( p^{32} T^{6} + p^{64} T^{8} \)
73$D_{4}$ \( ( 1 - 158814443769988 T + \)\(37\!\cdots\!62\)\( T^{2} - 158814443769988 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 - 2397008582592460 T + \)\(59\!\cdots\!22\)\( p T^{2} - 2397008582592460 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
83$C_2^2 \wr C_2$ \( 1 - \)\(46\!\cdots\!60\)\( T^{2} - \)\(18\!\cdots\!42\)\( T^{4} - \)\(46\!\cdots\!60\)\( p^{32} T^{6} + p^{64} T^{8} \)
89$C_2^2 \wr C_2$ \( 1 - \)\(45\!\cdots\!84\)\( T^{2} + \)\(22\!\cdots\!06\)\( T^{4} - \)\(45\!\cdots\!84\)\( p^{32} T^{6} + p^{64} T^{8} \)
97$D_{4}$ \( ( 1 + 6916897501300892 T + \)\(13\!\cdots\!62\)\( T^{2} + 6916897501300892 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−16.38522057107798428245705012450, −16.37787049749053713916100802538, −15.20664748669160562869486037550, −15.14068280125796332056697862769, −15.04846567527132458154960033715, −14.12165222353993348217533209508, −13.26976432337153163029235510906, −12.64276244970839965742851531576, −12.42351768649475129769376715329, −11.71164296627906324522090579035, −11.31120669693049391981952110807, −10.59660824468351979174914713129, −10.27199668072309756602977930882, −9.641806859839399850193019481738, −8.558340964066520449583168421918, −8.000104043634496999618136882987, −6.84783688008767126190920179119, −6.56313577358796467533377958539, −6.53455730748199774289560544157, −4.99031573645298447056423255859, −4.40448525074930780880881606505, −2.90750237650572410083716922065, −2.41699071672461749929038480110, −1.83488451980409632271054600575, −0.42218955733838694584384415923, 0.42218955733838694584384415923, 1.83488451980409632271054600575, 2.41699071672461749929038480110, 2.90750237650572410083716922065, 4.40448525074930780880881606505, 4.99031573645298447056423255859, 6.53455730748199774289560544157, 6.56313577358796467533377958539, 6.84783688008767126190920179119, 8.000104043634496999618136882987, 8.558340964066520449583168421918, 9.641806859839399850193019481738, 10.27199668072309756602977930882, 10.59660824468351979174914713129, 11.31120669693049391981952110807, 11.71164296627906324522090579035, 12.42351768649475129769376715329, 12.64276244970839965742851531576, 13.26976432337153163029235510906, 14.12165222353993348217533209508, 15.04846567527132458154960033715, 15.14068280125796332056697862769, 15.20664748669160562869486037550, 16.37787049749053713916100802538, 16.38522057107798428245705012450

Graph of the $Z$-function along the critical line