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Properties

Label 4-290e2-1.1-c1e2-0-1
Degree $4$
Conductor $84100$
Sign $1$
Analytic cond. $5.36228$
Root an. cond. $1.52172$
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  − 4-s + 4·5-s − 5·9-s − 5·11-s + 16-s + 10·19-s − 4·20-s + 11·25-s + 10·31-s + 5·36-s + 5·41-s + 5·44-s − 20·45-s + 10·49-s − 20·55-s + 10·59-s − 8·61-s − 64-s + 2·71-s − 10·76-s + 4·80-s + 16·81-s − 5·89-s + 40·95-s + 25·99-s − 11·100-s + 12·101-s + ⋯
L(s)  = 1  − 1/2·4-s + 1.78·5-s − 5/3·9-s − 1.50·11-s + 1/4·16-s + 2.29·19-s − 0.894·20-s + 11/5·25-s + 1.79·31-s + 5/6·36-s + 0.780·41-s + 0.753·44-s − 2.98·45-s + 10/7·49-s − 2.69·55-s + 1.30·59-s − 1.02·61-s − 1/8·64-s + 0.237·71-s − 1.14·76-s + 0.447·80-s + 16/9·81-s − 0.529·89-s + 4.10·95-s + 2.51·99-s − 1.09·100-s + 1.19·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(84100\)    =    \(2^{2} \cdot 5^{2} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(5.36228\)
Root analytic conductor: \(1.52172\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 84100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.554812297\)
\(L(\frac12)\) \(\approx\) \(1.554812297\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 - 4 T + p T^{2} \)
29$C_2$ \( 1 + p T^{2} \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.3.a_f
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
11$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.f_ba
13$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \) 2.13.a_p
17$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.17.a_ak
19$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.19.ak_cl
23$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.23.a_az
31$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.31.ak_di
37$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.37.a_z
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.41.af_cy
43$C_2^2$ \( 1 - 55 T^{2} + p^{2} T^{4} \) 2.43.a_acd
47$C_2^2$ \( 1 + 25 T^{2} + p^{2} T^{4} \) 2.47.a_z
53$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.53.a_az
59$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.59.ak_fn
61$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.61.i_cw
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.67.a_ak
71$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.71.ac_eo
73$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.73.a_be
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.79.a_cg
83$C_2^2$ \( 1 + 155 T^{2} + p^{2} T^{4} \) 2.83.a_fz
89$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.89.f_ey
97$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \) 2.97.a_adh
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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

−9.769260786131724864590575181723, −9.207099266666014523257104263415, −8.839944854222059986609889934704, −8.272327495974107231607593238799, −7.80808670813369143934158165671, −7.22663575360167386158402337109, −6.41798160914046102654872348906, −5.87559679810934793837804944661, −5.43789228568670930776406533980, −5.27941206710047657502553017219, −4.59654427537884444565275885142, −3.35223258618962479012226682226, −2.74373581426888450349801093293, −2.41508189098908449523740698579, −0.990439021529963904035976893885, 0.990439021529963904035976893885, 2.41508189098908449523740698579, 2.74373581426888450349801093293, 3.35223258618962479012226682226, 4.59654427537884444565275885142, 5.27941206710047657502553017219, 5.43789228568670930776406533980, 5.87559679810934793837804944661, 6.41798160914046102654872348906, 7.22663575360167386158402337109, 7.80808670813369143934158165671, 8.272327495974107231607593238799, 8.839944854222059986609889934704, 9.207099266666014523257104263415, 9.769260786131724864590575181723

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