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Properties

Label 4-290e2-1.1-c1e2-0-21
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 5-s − 2·9-s − 11-s + 16-s − 7·19-s − 20-s − 4·25-s − 7·29-s − 31-s + 2·36-s + 41-s + 44-s − 2·45-s − 49-s − 55-s − 4·61-s − 64-s − 71-s + 7·76-s − 79-s + 80-s − 5·81-s − 17·89-s − 7·95-s + 2·99-s + 4·100-s + ⋯
L(s)  = 1  − 1/2·4-s + 0.447·5-s − 2/3·9-s − 0.301·11-s + 1/4·16-s − 1.60·19-s − 0.223·20-s − 4/5·25-s − 1.29·29-s − 0.179·31-s + 1/3·36-s + 0.156·41-s + 0.150·44-s − 0.298·45-s − 1/7·49-s − 0.134·55-s − 0.512·61-s − 1/8·64-s − 0.118·71-s + 0.802·76-s − 0.112·79-s + 0.111·80-s − 5/9·81-s − 1.80·89-s − 0.718·95-s + 0.201·99-s + 2/5·100-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: \(1\)
Selberg data: \((4,\ 84100,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 - T + p T^{2} \)
29$C_2$ \( 1 + 7 T + p T^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
7$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.7.a_b
11$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.b_ai
13$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \) 2.13.a_ar
17$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.17.a_e
19$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.h_by
23$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.23.a_o
31$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.31.b_ci
37$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \) 2.37.a_bg
41$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.41.ab_ai
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.43.a_ao
47$C_2^2$ \( 1 + 60 T^{2} + p^{2} T^{4} \) 2.47.a_ci
53$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.53.a_aw
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.59.a_de
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.61.e_eg
67$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.67.a_cb
71$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.71.b_cs
73$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \) 2.73.a_abo
79$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.79.b_ba
83$C_2^2$ \( 1 + 97 T^{2} + p^{2} T^{4} \) 2.83.a_dt
89$C_2$$\times$$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.89.r_je
97$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.97.a_acw
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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

−9.407013177961134354514557583663, −8.996100385808886052912111582414, −8.473825051965990185415913679339, −8.071522902280307190846952485219, −7.52700228462482304525883443890, −6.85205748902706174509852826637, −6.28424295075064864128230587879, −5.64424982762544943495731673496, −5.47383466138333082739933250768, −4.52433454980790971630007774325, −4.09997858337187111625902353483, −3.31670759299562002360105872786, −2.48240814347726224049513869745, −1.73603270615453007779536366765, 0, 1.73603270615453007779536366765, 2.48240814347726224049513869745, 3.31670759299562002360105872786, 4.09997858337187111625902353483, 4.52433454980790971630007774325, 5.47383466138333082739933250768, 5.64424982762544943495731673496, 6.28424295075064864128230587879, 6.85205748902706174509852826637, 7.52700228462482304525883443890, 8.071522902280307190846952485219, 8.473825051965990185415913679339, 8.996100385808886052912111582414, 9.407013177961134354514557583663

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