Properties

Degree 4
Conductor $ 2^{2} \cdot 3^{2} \cdot 59^{2} $
Sign $1$
Motivic weight 3
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 6·3-s + 12·4-s + 26·5-s − 24·6-s − 32·8-s + 27·9-s − 104·10-s + 52·11-s + 72·12-s + 66·13-s + 156·15-s + 80·16-s − 64·17-s − 108·18-s + 68·19-s + 312·20-s − 208·22-s + 116·23-s − 192·24-s + 308·25-s − 264·26-s + 108·27-s + 350·29-s − 624·30-s − 242·31-s − 192·32-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 3/2·4-s + 2.32·5-s − 1.63·6-s − 1.41·8-s + 9-s − 3.28·10-s + 1.42·11-s + 1.73·12-s + 1.40·13-s + 2.68·15-s + 5/4·16-s − 0.913·17-s − 1.41·18-s + 0.821·19-s + 3.48·20-s − 2.01·22-s + 1.05·23-s − 1.63·24-s + 2.46·25-s − 1.99·26-s + 0.769·27-s + 2.24·29-s − 3.79·30-s − 1.40·31-s − 1.06·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 125316 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125316 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(125316\)    =    \(2^{2} \cdot 3^{2} \cdot 59^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  induced by $\chi_{354} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 125316,\ (\ :3/2, 3/2),\ 1)$
$L(2)$  $\approx$  $5.055880693$
$L(\frac12)$  $\approx$  $5.055880693$
$L(\frac{5}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;59\}$,\(F_p(T)\) is a polynomial of degree 4. If $p \in \{2,\;3,\;59\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 + p T )^{2} \)
3$C_1$ \( ( 1 - p T )^{2} \)
59$C_1$ \( ( 1 - p T )^{2} \)
good5$D_{4}$ \( 1 - 26 T + 368 T^{2} - 26 p^{3} T^{3} + p^{6} T^{4} \)
7$C_2^2$ \( 1 - 130 T^{2} + p^{6} T^{4} \)
11$D_{4}$ \( 1 - 52 T + 2522 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 - 66 T + 2984 T^{2} - 66 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 64 T + 9014 T^{2} + 64 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2$ \( ( 1 - 34 T + p^{3} T^{2} )^{2} \)
23$D_{4}$ \( 1 - 116 T + 25862 T^{2} - 116 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 - 350 T + 75272 T^{2} - 350 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 242 T + 59484 T^{2} + 242 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 638 T + 200568 T^{2} + 638 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 616 T + 216182 T^{2} + 616 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 768 T + 303206 T^{2} - 768 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 44 T + 134486 T^{2} - 44 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 90 T + 295648 T^{2} - 90 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 - 1250 T + 817608 T^{2} - 1250 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 1460 T + 1124430 T^{2} - 1460 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 162 T - 100196 T^{2} - 162 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 284 T + 203334 T^{2} - 284 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 512 T + 108318 T^{2} + 512 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 376 T + 708902 T^{2} - 376 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 492 T + 1057354 T^{2} + 492 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 1088 T + 1259382 T^{2} + 1088 p^{3} T^{3} + p^{6} T^{4} \)
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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

−11.00478231461511973422692794527, −10.52877254078328178390087423128, −10.11200340950897202232202702423, −9.817152716644443673319396659264, −9.322543066975650477180345815801, −8.824779134648160846813919739788, −8.735091791473356010379578870730, −8.537506757522985625002198102990, −7.52226873176453074277556292778, −6.88216624453555474779064443854, −6.60395135955651265288225434024, −6.37215331837829028975083721783, −5.43326695718227521189907725024, −5.16693076999975991205616861697, −3.80126073757737553701627632781, −3.48927258245031158604539977496, −2.42275211842072018208899188403, −2.17445645535342592666006834043, −1.26708412265135184728524806385, −1.15937265681046404676898343652, 1.15937265681046404676898343652, 1.26708412265135184728524806385, 2.17445645535342592666006834043, 2.42275211842072018208899188403, 3.48927258245031158604539977496, 3.80126073757737553701627632781, 5.16693076999975991205616861697, 5.43326695718227521189907725024, 6.37215331837829028975083721783, 6.60395135955651265288225434024, 6.88216624453555474779064443854, 7.52226873176453074277556292778, 8.537506757522985625002198102990, 8.735091791473356010379578870730, 8.824779134648160846813919739788, 9.322543066975650477180345815801, 9.817152716644443673319396659264, 10.11200340950897202232202702423, 10.52877254078328178390087423128, 11.00478231461511973422692794527

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