Properties

Degree $4$
Conductor $5531904$
Sign $1$
Motivic weight $3$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·3-s − 12·5-s + 27·9-s − 4·11-s − 48·13-s − 72·15-s − 132·17-s + 120·19-s + 76·23-s − 140·25-s + 108·27-s − 112·29-s + 432·31-s − 24·33-s − 280·37-s − 288·39-s − 36·41-s + 128·43-s − 324·45-s − 264·47-s − 792·51-s + 268·53-s + 48·55-s + 720·57-s + 336·59-s + 504·61-s + 576·65-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.07·5-s + 9-s − 0.109·11-s − 1.02·13-s − 1.23·15-s − 1.88·17-s + 1.44·19-s + 0.689·23-s − 1.11·25-s + 0.769·27-s − 0.717·29-s + 2.50·31-s − 0.126·33-s − 1.24·37-s − 1.18·39-s − 0.137·41-s + 0.453·43-s − 1.07·45-s − 0.819·47-s − 2.17·51-s + 0.694·53-s + 0.117·55-s + 1.67·57-s + 0.741·59-s + 1.05·61-s + 1.09·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5531904\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(3\)
Character: induced by $\chi_{2352} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5531904,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(4.207991965\)
\(L(\frac12)\) \(\approx\) \(4.207991965\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_1$ \( ( 1 - p T )^{2} \)
7 \( 1 \)
good5$D_{4}$ \( 1 + 12 T + 284 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 + 4 T + 2594 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 48 T + 4520 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 132 T + 820 p T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 - 120 T + 13086 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 - 76 T + 4970 T^{2} - 76 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 112 T + 6914 T^{2} + 112 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 432 T + 100406 T^{2} - 432 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 280 T + 56106 T^{2} + 280 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 36 T + 105908 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 128 T - 2778 T^{2} - 128 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 264 T - 3418 T^{2} + 264 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 268 T + 241982 T^{2} - 268 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 336 T + 261374 T^{2} - 336 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 - 504 T + 268248 T^{2} - 504 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 384 T + 596918 T^{2} - 384 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 396 T + 521098 T^{2} - 396 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 312 T + 94320 T^{2} - 312 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 848 T + 985854 T^{2} - 848 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 648 T + 1235750 T^{2} + 648 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 612 T + 1442324 T^{2} - 612 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 - 2184 T + 2982432 T^{2} - 2184 p^{3} T^{3} + p^{6} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.633317885479656665747045176627, −8.464310852592366618091940798794, −8.034754157404935404986058702397, −7.78359990256618150708105149294, −7.33111070285033705664592543350, −7.07781567827630552808467759151, −6.62481349207779177984000378437, −6.37655145753054189368115001657, −5.50245667642303507380810097949, −5.20290814148075528968259055727, −4.73823510772649178941135367092, −4.30305942888510030851443219707, −3.98662607110675946174460996874, −3.50505978795658088630707467199, −3.05398772055304615381578360733, −2.65217991889846832478146604881, −2.04550379913908738161928337724, −1.82449412407694517841270931693, −0.65023772706079081117068414162, −0.56710428894951888202403424681, 0.56710428894951888202403424681, 0.65023772706079081117068414162, 1.82449412407694517841270931693, 2.04550379913908738161928337724, 2.65217991889846832478146604881, 3.05398772055304615381578360733, 3.50505978795658088630707467199, 3.98662607110675946174460996874, 4.30305942888510030851443219707, 4.73823510772649178941135367092, 5.20290814148075528968259055727, 5.50245667642303507380810097949, 6.37655145753054189368115001657, 6.62481349207779177984000378437, 7.07781567827630552808467759151, 7.33111070285033705664592543350, 7.78359990256618150708105149294, 8.034754157404935404986058702397, 8.464310852592366618091940798794, 8.633317885479656665747045176627

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