Properties

Degree 4
Conductor $ 3^{2} \cdot 5^{2} \cdot 7^{2} $
Sign $1$
Motivic weight 3
Primitive no
Self-dual yes
Analytic rank 2

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 6·3-s + 4-s − 10·5-s − 24·6-s − 14·7-s + 24·8-s + 27·9-s + 40·10-s − 92·11-s + 6·12-s + 8·13-s + 56·14-s − 60·15-s − 47·16-s − 44·17-s − 108·18-s − 108·19-s − 10·20-s − 84·21-s + 368·22-s − 320·23-s + 144·24-s + 75·25-s − 32·26-s + 108·27-s − 14·28-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 1/8·4-s − 0.894·5-s − 1.63·6-s − 0.755·7-s + 1.06·8-s + 9-s + 1.26·10-s − 2.52·11-s + 0.144·12-s + 0.170·13-s + 1.06·14-s − 1.03·15-s − 0.734·16-s − 0.627·17-s − 1.41·18-s − 1.30·19-s − 0.111·20-s − 0.872·21-s + 3.56·22-s − 2.90·23-s + 1.22·24-s + 3/5·25-s − 0.241·26-s + 0.769·27-s − 0.0944·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(11025\)    =    \(3^{2} \cdot 5^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  induced by $\chi_{105} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(2\)
Selberg data  =  \((4,\ 11025,\ (\ :3/2, 3/2),\ 1)\)
\(L(2)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{5}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 4. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$ \( ( 1 - p T )^{2} \)
5$C_1$ \( ( 1 + p T )^{2} \)
7$C_1$ \( ( 1 + p T )^{2} \)
good2$D_{4}$ \( 1 + p^{2} T + 15 T^{2} + p^{5} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 + 92 T + 4758 T^{2} + 92 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 - 8 T - 2810 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 44 T + 630 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 + 108 T + 13254 T^{2} + 108 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 320 T + 47934 T^{2} + 320 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 236 T + 61982 T^{2} + 236 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 60 T + 8462 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 - 204 T + 108830 T^{2} - 204 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 44 T + 106326 T^{2} - 44 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 136 T + 81718 T^{2} - 136 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 400 T + 106526 T^{2} - 400 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 16 T + 260838 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 464 T + 458102 T^{2} + 464 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 684 T + 535646 T^{2} + 684 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 736 T + 602470 T^{2} - 736 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 740 T + 757502 T^{2} + 740 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 424 T + 748558 T^{2} - 424 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 408 T - 143586 T^{2} + 408 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 608 T + 1200710 T^{2} - 608 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 1332 T + 1790774 T^{2} + 1332 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 2448 T + 3286542 T^{2} + 2448 p^{3} T^{3} + p^{6} T^{4} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−12.88181569707492296853043281622, −12.78558300354075809399407693352, −12.09629816446006541908408907175, −11.02195019693420594559702105497, −10.65386272950849421210188056087, −10.16767996106242022361660541788, −9.684910701443008864898681857926, −9.168981172629181272824934474395, −8.535523732055619904725233737761, −8.182929901094436683442313395985, −7.62799052378416580530155802593, −7.56005056030982890340253285064, −6.33003777724750842570572638412, −5.51723347212477855927934184572, −4.29591832719651791160000479572, −4.01589893874121749634155343159, −2.81449756126596726250098903133, −2.09774235159822270881569792749, 0, 0, 2.09774235159822270881569792749, 2.81449756126596726250098903133, 4.01589893874121749634155343159, 4.29591832719651791160000479572, 5.51723347212477855927934184572, 6.33003777724750842570572638412, 7.56005056030982890340253285064, 7.62799052378416580530155802593, 8.182929901094436683442313395985, 8.535523732055619904725233737761, 9.168981172629181272824934474395, 9.684910701443008864898681857926, 10.16767996106242022361660541788, 10.65386272950849421210188056087, 11.02195019693420594559702105497, 12.09629816446006541908408907175, 12.78558300354075809399407693352, 12.88181569707492296853043281622

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