Properties

Label 4-1040e2-1.1-c1e2-0-42
Degree $4$
Conductor $1081600$
Sign $1$
Analytic cond. $68.9637$
Root an. cond. $2.88174$
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  + 2·5-s − 2·9-s + 2·13-s + 4·17-s − 25-s + 12·29-s + 12·37-s + 4·41-s − 4·45-s + 6·49-s − 8·53-s + 4·61-s + 4·65-s − 4·73-s − 5·81-s + 8·85-s + 24·89-s − 16·97-s + 12·101-s − 20·109-s + 12·113-s − 4·117-s + 14·121-s − 12·125-s + ⋯
L(s)  = 1  + 0.894·5-s − 2/3·9-s + 0.554·13-s + 0.970·17-s − 1/5·25-s + 2.22·29-s + 1.97·37-s + 0.624·41-s − 0.596·45-s + 6/7·49-s − 1.09·53-s + 0.512·61-s + 0.496·65-s − 0.468·73-s − 5/9·81-s + 0.867·85-s + 2.54·89-s − 1.62·97-s + 1.19·101-s − 1.91·109-s + 1.12·113-s − 0.369·117-s + 1.27·121-s − 1.07·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1081600\)    =    \(2^{8} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(68.9637\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1081600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.726198904\)
\(L(\frac12)\) \(\approx\) \(2.726198904\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
13$C_2$ \( 1 - 2 T + p T^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
7$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.7.a_ag
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.a_ao
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.ae_w
19$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.19.a_ba
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.23.a_s
29$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.29.am_da
31$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.31.a_aw
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.37.am_dq
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.ae_cs
43$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.43.a_ag
47$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.47.a_k
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.53.i_w
59$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.59.a_cg
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.61.ae_ck
67$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \) 2.67.a_acc
71$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.71.a_abm
73$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.e_di
79$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.79.a_ac
83$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.83.a_abu
89$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) 2.89.ay_mg
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.q_io
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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

−8.141849759960894334677229027623, −7.59363271476802522338037058465, −7.37756556473748469033529732660, −6.44058657537095987777695401445, −6.32481442184595602834156885923, −5.97838962103619549952431278213, −5.46821532544896628058214429870, −5.01134927167990789290746524605, −4.48074509022060414009145827128, −3.93006648049942723122764370930, −3.28653389892603790705696771040, −2.71047174767750097979379161069, −2.37937687292681073111960776302, −1.42436110620876358464543868604, −0.816904815074935296548574450648, 0.816904815074935296548574450648, 1.42436110620876358464543868604, 2.37937687292681073111960776302, 2.71047174767750097979379161069, 3.28653389892603790705696771040, 3.93006648049942723122764370930, 4.48074509022060414009145827128, 5.01134927167990789290746524605, 5.46821532544896628058214429870, 5.97838962103619549952431278213, 6.32481442184595602834156885923, 6.44058657537095987777695401445, 7.37756556473748469033529732660, 7.59363271476802522338037058465, 8.141849759960894334677229027623

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