Properties

Label 4-2352e2-1.1-c3e2-0-11
Degree $4$
Conductor $5531904$
Sign $1$
Analytic cond. $19257.8$
Root an. cond. $11.7801$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·3-s − 9·5-s + 27·9-s − 5·11-s + 19·13-s + 54·15-s + 4·17-s + 117·19-s − 148·23-s − 63·25-s − 108·27-s − 95·29-s − 360·31-s + 30·33-s − 53·37-s − 114·39-s + 170·41-s − 403·43-s − 243·45-s + 368·47-s − 24·51-s + 697·53-s + 45·55-s − 702·57-s − 585·59-s + 1.16e3·61-s − 171·65-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.804·5-s + 9-s − 0.137·11-s + 0.405·13-s + 0.929·15-s + 0.0570·17-s + 1.41·19-s − 1.34·23-s − 0.503·25-s − 0.769·27-s − 0.608·29-s − 2.08·31-s + 0.158·33-s − 0.235·37-s − 0.468·39-s + 0.647·41-s − 1.42·43-s − 0.804·45-s + 1.14·47-s − 0.0658·51-s + 1.80·53-s + 0.110·55-s − 1.63·57-s − 1.29·59-s + 2.43·61-s − 0.326·65-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: $1$
Analytic conductor: \(19257.8\)
Root analytic conductor: \(11.7801\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 5531904,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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$D_{4}$ \( 1 + 9 T + 144 T^{2} + 9 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 + 5 T - 488 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 - 19 T + 4358 T^{2} - 19 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 - 4 T + 7810 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 - 117 T + 10954 T^{2} - 117 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 148 T + 27790 T^{2} + 148 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 95 T + 47878 T^{2} + 95 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 360 T + 91477 T^{2} + 360 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 53 T + 101882 T^{2} + 53 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 170 T + 104162 T^{2} - 170 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 403 T + 107580 T^{2} + 403 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 368 T + 77882 T^{2} - 368 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 697 T + 7850 p T^{2} - 697 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 585 T + 404278 T^{2} + 585 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 - 1160 T + 691382 T^{2} - 1160 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 233 T - 57688 T^{2} + 233 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 616 T + 81466 T^{2} + 616 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 817 T + 443820 T^{2} - 817 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 802 T + 855999 T^{2} - 802 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 283 T + 596860 T^{2} + 283 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 1858 T + 2211874 T^{2} - 1858 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 1729 T + 2190800 T^{2} + 1729 p^{3} T^{3} + p^{6} T^{4} \)
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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

−8.232402943816877004546267887345, −8.084294683307314176167441609457, −7.46705679693350887609067261256, −7.38605384484623317024529394333, −6.96273980807054071107151762700, −6.49949924546603213422567137501, −5.98601650598892647282687477334, −5.65176391001773786413303138164, −5.23391041393623504303682546275, −5.19265937773197654925287430332, −4.17419845927221515075399876332, −4.17223304841035422242391495560, −3.49025522646546413279986908319, −3.46292562163118830031119333370, −2.40004342483679636100932789409, −2.02701624497428652809097063538, −1.32164763065636438092304629326, −0.864298108556478629796836981150, 0, 0, 0.864298108556478629796836981150, 1.32164763065636438092304629326, 2.02701624497428652809097063538, 2.40004342483679636100932789409, 3.46292562163118830031119333370, 3.49025522646546413279986908319, 4.17223304841035422242391495560, 4.17419845927221515075399876332, 5.19265937773197654925287430332, 5.23391041393623504303682546275, 5.65176391001773786413303138164, 5.98601650598892647282687477334, 6.49949924546603213422567137501, 6.96273980807054071107151762700, 7.38605384484623317024529394333, 7.46705679693350887609067261256, 8.084294683307314176167441609457, 8.232402943816877004546267887345

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