Properties

Label 4-1216e2-1.1-c1e2-0-22
Degree $4$
Conductor $1478656$
Sign $-1$
Analytic cond. $94.2803$
Root an. cond. $3.11605$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9-s + 4·13-s − 6·17-s − 6·25-s + 12·29-s + 8·37-s − 5·49-s − 12·53-s − 8·61-s − 2·73-s − 8·81-s − 12·89-s − 12·97-s − 12·101-s − 12·109-s + 12·113-s + 4·117-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 6·153-s + 157-s + 163-s + ⋯
L(s)  = 1  + 1/3·9-s + 1.10·13-s − 1.45·17-s − 6/5·25-s + 2.22·29-s + 1.31·37-s − 5/7·49-s − 1.64·53-s − 1.02·61-s − 0.234·73-s − 8/9·81-s − 1.27·89-s − 1.21·97-s − 1.19·101-s − 1.14·109-s + 1.12·113-s + 0.369·117-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.485·153-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1478656\)    =    \(2^{12} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(94.2803\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1478656} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1478656,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 27 T^{2} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 3 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 57 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
79$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \)
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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

−7.79479479553236507364486493732, −7.27774099213358132015809486803, −6.68957202721806193978273217581, −6.35014987329731172636910476357, −6.18099911021454210276504007249, −5.58353039983297327223378157199, −4.90711614985539699610999865212, −4.52831809305135001637386747581, −4.13546939951366041525236225079, −3.68248419208624114766605593517, −2.86217733411437857444907232853, −2.60478905040558338261948072875, −1.68216410306367556145215742738, −1.19792738995265454495203065562, 0, 1.19792738995265454495203065562, 1.68216410306367556145215742738, 2.60478905040558338261948072875, 2.86217733411437857444907232853, 3.68248419208624114766605593517, 4.13546939951366041525236225079, 4.52831809305135001637386747581, 4.90711614985539699610999865212, 5.58353039983297327223378157199, 6.18099911021454210276504007249, 6.35014987329731172636910476357, 6.68957202721806193978273217581, 7.27774099213358132015809486803, 7.79479479553236507364486493732

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