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Properties

Label 4-10400-1.1-c1e2-0-0
Degree $4$
Conductor $10400$
Sign $1$
Analytic cond. $0.663113$
Root an. cond. $0.902395$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 8-s − 2·9-s + 3·13-s + 16-s − 2·18-s + 25-s + 3·26-s + 32-s − 2·36-s − 20·37-s − 12·41-s + 2·49-s + 50-s + 3·52-s + 12·53-s − 8·61-s + 64-s − 2·72-s + 4·73-s − 20·74-s − 5·81-s − 12·82-s + 4·97-s + 2·98-s + 100-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s − 2/3·9-s + 0.832·13-s + 1/4·16-s − 0.471·18-s + 1/5·25-s + 0.588·26-s + 0.176·32-s − 1/3·36-s − 3.28·37-s − 1.87·41-s + 2/7·49-s + 0.141·50-s + 0.416·52-s + 1.64·53-s − 1.02·61-s + 1/8·64-s − 0.235·72-s + 0.468·73-s − 2.32·74-s − 5/9·81-s − 1.32·82-s + 0.406·97-s + 0.202·98-s + 1/10·100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10400\)    =    \(2^{5} \cdot 5^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(0.663113\)
Root analytic conductor: \(0.902395\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.446424644\)
\(L(\frac12)\) \(\approx\) \(1.446424644\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
13$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 2 T + p T^{2} ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.11.a_k
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.19.a_ao
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.23.a_ac
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.37.u_gs
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.43.a_bi
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.a_aby
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \) 2.59.a_ack
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.i_dy
67$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.67.a_w
71$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.71.a_aba
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.73.ae_fu
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.79.a_aby
83$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.83.a_aby
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.a_fm
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.ae_cc
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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

−11.66549355791114539385425625313, −10.96524464611019582571978985532, −10.46704351419764864999841421058, −10.07263440267819806595627657880, −9.033955783317675403533705613851, −8.645043036008213773057209222665, −8.150645001021865976615651167824, −7.14503475842173199005946453357, −6.80751253979287515069378832502, −5.95218615347189775520974666441, −5.41744248357718446106164826189, −4.74828750156936176363468698131, −3.69549092796002537459428391596, −3.18191093703696527759015834206, −1.88204151833248830237718914676, 1.88204151833248830237718914676, 3.18191093703696527759015834206, 3.69549092796002537459428391596, 4.74828750156936176363468698131, 5.41744248357718446106164826189, 5.95218615347189775520974666441, 6.80751253979287515069378832502, 7.14503475842173199005946453357, 8.150645001021865976615651167824, 8.645043036008213773057209222665, 9.033955783317675403533705613851, 10.07263440267819806595627657880, 10.46704351419764864999841421058, 10.96524464611019582571978985532, 11.66549355791114539385425625313

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