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Properties

Label 4-810e2-1.1-c1e2-0-11
Degree $4$
Conductor $656100$
Sign $1$
Analytic cond. $41.8335$
Root an. cond. $2.54320$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 5-s − 2·7-s + 8-s + 10-s + 3·11-s + 13-s + 2·14-s − 16-s − 6·17-s + 16·19-s − 3·22-s − 3·23-s − 26-s + 9·29-s + 7·31-s + 6·34-s + 2·35-s + 4·37-s − 16·38-s − 40-s + 12·41-s + 7·43-s + 3·46-s + 3·47-s + 7·49-s + 24·53-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.447·5-s − 0.755·7-s + 0.353·8-s + 0.316·10-s + 0.904·11-s + 0.277·13-s + 0.534·14-s − 1/4·16-s − 1.45·17-s + 3.67·19-s − 0.639·22-s − 0.625·23-s − 0.196·26-s + 1.67·29-s + 1.25·31-s + 1.02·34-s + 0.338·35-s + 0.657·37-s − 2.59·38-s − 0.158·40-s + 1.87·41-s + 1.06·43-s + 0.442·46-s + 0.437·47-s + 49-s + 3.29·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(656100\)    =    \(2^{2} \cdot 3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(41.8335\)
Root analytic conductor: \(2.54320\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 656100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.394376094\)
\(L(\frac12)\) \(\approx\) \(1.394376094\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
3 \( 1 \)
5$C_2$ \( 1 + T + T^{2} \)
good7$C_2^2$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_ad
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.11.ad_ac
13$C_2^2$ \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) 2.13.ab_am
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.17.g_br
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.19.aq_dy
23$C_2^2$ \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.23.d_ao
29$C_2^2$ \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.29.aj_ca
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.ah_s
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2^2$ \( 1 - 12 T + 103 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.41.am_dz
43$C_2^2$ \( 1 - 7 T + 6 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.43.ah_g
47$C_2^2$ \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.47.ad_abm
53$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.53.ay_jq
59$C_2^2$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.59.m_dh
61$C_2^2$ \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.61.ak_bn
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_abz
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.73.ae_fu
79$C_2^2$ \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) 2.79.ab_ada
83$C_2^2$ \( 1 + 18 T + 241 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.83.s_jh
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) 2.97.o_dv
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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

−10.17297224337387944583118350149, −9.990634363479202083667062425642, −9.552780408871169317454322629247, −9.306865214777139839220019479738, −8.697107750584296579505238978685, −8.585645933276690622305387548347, −7.890592410807507198691274491193, −7.42023217773605918643018096829, −7.15535967948534835124042403873, −6.74762953638679872810675554586, −6.04101124331333420337436293070, −5.82591596801831856822875050826, −5.12936244684128544566725225580, −4.49649915728998358895675388480, −4.02248190083149546310194743341, −3.64841607276553205978961270785, −2.69559313048512516788048661835, −2.58986031721244231019437286714, −1.04524584204665034845517802725, −0.907464480540244072509983894920, 0.907464480540244072509983894920, 1.04524584204665034845517802725, 2.58986031721244231019437286714, 2.69559313048512516788048661835, 3.64841607276553205978961270785, 4.02248190083149546310194743341, 4.49649915728998358895675388480, 5.12936244684128544566725225580, 5.82591596801831856822875050826, 6.04101124331333420337436293070, 6.74762953638679872810675554586, 7.15535967948534835124042403873, 7.42023217773605918643018096829, 7.890592410807507198691274491193, 8.585645933276690622305387548347, 8.697107750584296579505238978685, 9.306865214777139839220019479738, 9.552780408871169317454322629247, 9.990634363479202083667062425642, 10.17297224337387944583118350149

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