Properties

Label 4-124e2-1.1-c3e2-0-0
Degree $4$
Conductor $15376$
Sign $1$
Analytic cond. $53.5273$
Root an. cond. $2.70485$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 7·4-s − 4·5-s − 15·8-s − 54·9-s − 4·10-s + 41·16-s − 54·18-s + 28·20-s − 238·25-s + 161·32-s + 378·36-s + 60·40-s − 556·41-s + 216·45-s − 430·49-s − 238·50-s − 167·64-s + 810·72-s − 164·80-s + 2.18e3·81-s − 556·82-s + 216·90-s + 3.81e3·97-s − 430·98-s + 1.66e3·100-s + 3.58e3·101-s + ⋯
L(s)  = 1  + 0.353·2-s − 7/8·4-s − 0.357·5-s − 0.662·8-s − 2·9-s − 0.126·10-s + 0.640·16-s − 0.707·18-s + 0.313·20-s − 1.90·25-s + 0.889·32-s + 7/4·36-s + 0.237·40-s − 2.11·41-s + 0.715·45-s − 1.25·49-s − 0.673·50-s − 0.326·64-s + 1.32·72-s − 0.229·80-s + 3·81-s − 0.748·82-s + 0.252·90-s + 3.99·97-s − 0.443·98-s + 1.66·100-s + 3.52·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15376 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(15376\)    =    \(2^{4} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(53.5273\)
Root analytic conductor: \(2.70485\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 15376,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5720767734\)
\(L(\frac12)\) \(\approx\) \(0.5720767734\)
\(L(\frac{5}{2})\) 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$C_2$ \( 1 - T + p^{3} T^{2} \)
31$C_2$ \( 1 + p^{3} T^{2} \)
good3$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
5$C_2$ \( ( 1 + 2 T + p^{3} T^{2} )^{2} \)
7$C_2$ \( ( 1 - 16 T + p^{3} T^{2} )( 1 + 16 T + p^{3} T^{2} ) \)
11$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
13$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
17$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
19$C_2$ \( ( 1 - 156 T + p^{3} T^{2} )( 1 + 156 T + p^{3} T^{2} ) \)
23$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
29$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
37$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
41$C_2$ \( ( 1 + 278 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
47$C_2$ \( ( 1 - 616 T + p^{3} T^{2} )( 1 + 616 T + p^{3} T^{2} ) \)
53$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
59$C_2$ \( ( 1 - 740 T + p^{3} T^{2} )( 1 + 740 T + p^{3} T^{2} ) \)
61$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
67$C_2$ \( ( 1 - 684 T + p^{3} T^{2} )( 1 + 684 T + p^{3} T^{2} ) \)
71$C_2$ \( ( 1 - 1000 T + p^{3} T^{2} )( 1 + 1000 T + p^{3} T^{2} ) \)
73$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
79$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
83$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
89$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 1906 T + p^{3} T^{2} )^{2} \)
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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

−13.24965469543222111953890737899, −12.81268344637674264412799311812, −11.92391377500608265827532219908, −11.74354484781918643953353949918, −11.45203557702265082189986049585, −10.58056503746539082675882648508, −10.05350103809888349052676677392, −9.394121043452456148714338570571, −8.923107405462924579964981770043, −8.291986757122217922277701682317, −8.098104056701677226481838952610, −7.29244739467627077681133455726, −6.15548640948323978985853591114, −6.00192615884594482724549271996, −5.16300131950600311541676092085, −4.69108481642176139809910298238, −3.54136928556152791095773913448, −3.33298338377701237097363509113, −2.12503024523908125777662830582, −0.36425813691943199516077323340, 0.36425813691943199516077323340, 2.12503024523908125777662830582, 3.33298338377701237097363509113, 3.54136928556152791095773913448, 4.69108481642176139809910298238, 5.16300131950600311541676092085, 6.00192615884594482724549271996, 6.15548640948323978985853591114, 7.29244739467627077681133455726, 8.098104056701677226481838952610, 8.291986757122217922277701682317, 8.923107405462924579964981770043, 9.394121043452456148714338570571, 10.05350103809888349052676677392, 10.58056503746539082675882648508, 11.45203557702265082189986049585, 11.74354484781918643953353949918, 11.92391377500608265827532219908, 12.81268344637674264412799311812, 13.24965469543222111953890737899

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