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Properties

Label 4-15392-1.1-c1e2-0-1
Degree $4$
Conductor $15392$
Sign $-1$
Analytic cond. $0.981407$
Root an. cond. $0.995319$
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  − 2-s + 4-s − 3·5-s − 8-s − 2·9-s + 3·10-s − 3·13-s + 16-s − 3·17-s + 2·18-s − 3·20-s − 25-s + 3·26-s − 12·29-s − 32-s + 3·34-s − 2·36-s + 12·37-s + 3·40-s − 3·41-s + 6·45-s − 4·49-s + 50-s − 3·52-s − 3·53-s + 12·58-s − 2·61-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 1.34·5-s − 0.353·8-s − 2/3·9-s + 0.948·10-s − 0.832·13-s + 1/4·16-s − 0.727·17-s + 0.471·18-s − 0.670·20-s − 1/5·25-s + 0.588·26-s − 2.22·29-s − 0.176·32-s + 0.514·34-s − 1/3·36-s + 1.97·37-s + 0.474·40-s − 0.468·41-s + 0.894·45-s − 4/7·49-s + 0.141·50-s − 0.416·52-s − 0.412·53-s + 1.57·58-s − 0.256·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15392 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15392 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(15392\)    =    \(2^{5} \cdot 13 \cdot 37\)
Sign: $-1$
Analytic conductor: \(0.981407\)
Root analytic conductor: \(0.995319\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 15392,\ (\ :1/2, 1/2),\ -1)\)

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)$Isogeny Class over $\mathbf{F}_p$
bad2$C_1$ \( 1 + T \)
13$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 4 T + p T^{2} ) \)
37$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 11 T + p T^{2} ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
5$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.d_k
7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.7.a_e
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.11.a_n
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.d_bi
19$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.19.a_au
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.29.m_dh
31$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \) 2.31.a_ar
41$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.41.d_aba
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.43.a_ao
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.a_cg
53$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.d_dk
59$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.59.a_k
61$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.c_bq
67$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.67.a_abj
71$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.71.a_bi
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.73.f_fc
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.79.a_cg
83$C_2^2$ \( 1 + 103 T^{2} + p^{2} T^{4} \) 2.83.a_dz
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.89.d_eu
97$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.c_ek
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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

−11.07099497143921567696736962533, −10.27758266540498857533535975434, −9.612842489937553979620501714436, −9.230629787241096442511201074656, −8.560273336305024828529510334638, −7.983743912092236285941687098530, −7.55921512598335596563165485045, −7.16148717241414954142687107794, −6.24870451633718126286875293540, −5.65547119685705140292709047071, −4.69573503908089616509307460531, −3.98845089645888067713438384652, −3.17056529271174275714948989557, −2.13250160684136193159915900504, 0, 2.13250160684136193159915900504, 3.17056529271174275714948989557, 3.98845089645888067713438384652, 4.69573503908089616509307460531, 5.65547119685705140292709047071, 6.24870451633718126286875293540, 7.16148717241414954142687107794, 7.55921512598335596563165485045, 7.983743912092236285941687098530, 8.560273336305024828529510334638, 9.230629787241096442511201074656, 9.612842489937553979620501714436, 10.27758266540498857533535975434, 11.07099497143921567696736962533

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