↑

Properties

Label 4-290e2-1.1-c1e2-0-28
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 4·5-s − 5·9-s − 10·13-s + 16-s − 4·20-s − 10·23-s + 11·25-s + 5·36-s − 20·45-s − 10·49-s + 10·52-s − 64-s − 40·65-s + 5·67-s + 16·71-s + 4·80-s + 16·81-s − 10·83-s + 10·92-s − 11·100-s − 10·103-s − 25·107-s − 10·109-s − 40·115-s + 50·117-s + 7·121-s + ⋯
L(s)  = 1  − 1/2·4-s + 1.78·5-s − 5/3·9-s − 2.77·13-s + 1/4·16-s − 0.894·20-s − 2.08·23-s + 11/5·25-s + 5/6·36-s − 2.98·45-s − 1.42·49-s + 1.38·52-s − 1/8·64-s − 4.96·65-s + 0.610·67-s + 1.89·71-s + 0.447·80-s + 16/9·81-s − 1.09·83-s + 1.04·92-s − 1.09·100-s − 0.985·103-s − 2.41·107-s − 0.957·109-s − 3.73·115-s + 4.62·117-s + 7/11·121-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$C_2$ \( 1 - 4 T + p T^{2} \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \) 2.11.a_ah
13$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.k_by
17$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \) 2.17.a_f
19$C_2^2$ \( 1 + 33 T^{2} + p^{2} T^{4} \) 2.19.a_bh
23$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.k_cs
31$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \) 2.31.a_ah
37$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.37.a_az
41$C_2^2$ \( 1 + 13 T^{2} + p^{2} T^{4} \) 2.41.a_n
43$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.43.a_k
47$C_2^2$ \( 1 + 65 T^{2} + p^{2} T^{4} \) 2.47.a_cn
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.59.a_dp
61$C_2^2$ \( 1 + 63 T^{2} + p^{2} T^{4} \) 2.61.a_cl
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.67.af_by
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.71.aq_hy
73$C_2^2$ \( 1 - 95 T^{2} + p^{2} T^{4} \) 2.73.a_adr
79$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \) 2.79.a_afm
83$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.83.k_gt
89$C_2^2$ \( 1 + 133 T^{2} + p^{2} T^{4} \) 2.89.a_fd
97$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.97.a_abe
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.493804712703749678348201421332, −9.235959970736215532760034425974, −8.430349725203158684666416485246, −8.086930599635240852691343397900, −7.52638108795412483040413831120, −6.68592794123519736081953898196, −6.34369928485286719550333205031, −5.60908329356259112732112409588, −5.32359357463748579963392022920, −4.93065124027851561623507226273, −4.09284638527273191171582365805, −2.97323080425201066628866120217, −2.48617695509459991461013286158, −1.93376116970253608477934786556, 0, 1.93376116970253608477934786556, 2.48617695509459991461013286158, 2.97323080425201066628866120217, 4.09284638527273191171582365805, 4.93065124027851561623507226273, 5.32359357463748579963392022920, 5.60908329356259112732112409588, 6.34369928485286719550333205031, 6.68592794123519736081953898196, 7.52638108795412483040413831120, 8.086930599635240852691343397900, 8.430349725203158684666416485246, 9.235959970736215532760034425974, 9.493804712703749678348201421332

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