Properties

Label 4-2352e2-1.1-c3e2-0-16
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·3-s − 5·5-s + 27·9-s − 67·11-s − 41·13-s − 30·15-s + 92·17-s + 43·19-s − 148·23-s + 105·25-s + 108·27-s + 77·29-s − 520·31-s − 402·33-s + 7·37-s − 246·39-s + 426·41-s + 107·43-s − 135·45-s + 576·47-s + 552·51-s − 243·53-s + 335·55-s + 258·57-s − 7·59-s − 224·61-s + 205·65-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.447·5-s + 9-s − 1.83·11-s − 0.874·13-s − 0.516·15-s + 1.31·17-s + 0.519·19-s − 1.34·23-s + 0.839·25-s + 0.769·27-s + 0.493·29-s − 3.01·31-s − 2.12·33-s + 0.0311·37-s − 1.01·39-s + 1.62·41-s + 0.379·43-s − 0.447·45-s + 1.78·47-s + 1.51·51-s − 0.629·53-s + 0.821·55-s + 0.599·57-s − 0.0154·59-s − 0.470·61-s + 0.391·65-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: induced by $\chi_{2352} (1, \cdot )$
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$D_{4}$ \( 1 + p T - 16 p T^{2} + p^{4} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 + 67 T + 3448 T^{2} + 67 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 41 T + 4478 T^{2} + 41 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 - 92 T + 386 p T^{2} - 92 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 - 43 T + 11154 T^{2} - 43 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 148 T + 24430 T^{2} + 148 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 - 77 T + 9574 T^{2} - 77 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 520 T + 125837 T^{2} + 520 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 - 7 T + 74082 T^{2} - 7 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 426 T + 171106 T^{2} - 426 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 107 T + 86220 T^{2} - 107 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 576 T + 242170 T^{2} - 576 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 243 T + 309490 T^{2} + 243 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 7 T + 200614 T^{2} + 7 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 224 T + 461126 T^{2} + 224 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 687 T + 678832 T^{2} + 687 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 472 T + 637018 T^{2} + 472 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 921 T + 578188 T^{2} - 921 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 526 T + 1033727 T^{2} - 526 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 221 T + 945628 T^{2} + 221 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 774 T + 966562 T^{2} + 774 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 1953 T + 2366992 T^{2} + 1953 p^{3} T^{3} + p^{6} T^{4} \)
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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

−8.199057350642650573762978748187, −8.078438837358060382094484112350, −7.58193314722444538517062981921, −7.57678933914051344944378316490, −7.15872249860703764316768439876, −6.74614040972023023945990184685, −5.84051949655477768653289217985, −5.71181474699281042245901602972, −5.27352194504465938286283802917, −4.90505587889398848558486199389, −4.27298837167959193524494303464, −3.90868692286935363576877822366, −3.52881319785242172628897112093, −2.92919899627255661567124204811, −2.52984359047893878765173537965, −2.39653278274862668719268326507, −1.53450241016768092346441125728, −1.07587806306217472055390805781, 0, 0, 1.07587806306217472055390805781, 1.53450241016768092346441125728, 2.39653278274862668719268326507, 2.52984359047893878765173537965, 2.92919899627255661567124204811, 3.52881319785242172628897112093, 3.90868692286935363576877822366, 4.27298837167959193524494303464, 4.90505587889398848558486199389, 5.27352194504465938286283802917, 5.71181474699281042245901602972, 5.84051949655477768653289217985, 6.74614040972023023945990184685, 7.15872249860703764316768439876, 7.57678933914051344944378316490, 7.58193314722444538517062981921, 8.078438837358060382094484112350, 8.199057350642650573762978748187

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