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Properties

Label 4-1792e2-1.1-c1e2-0-9
Degree $4$
Conductor $3211264$
Sign $1$
Analytic cond. $204.752$
Root an. cond. $3.78274$
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  + 2·3-s + 2·5-s − 2·7-s − 4·11-s + 10·13-s + 4·15-s + 4·17-s − 6·19-s − 4·21-s − 4·25-s − 2·27-s + 4·29-s + 8·31-s − 8·33-s − 4·35-s + 4·37-s + 20·39-s + 4·41-s − 4·43-s − 8·47-s + 3·49-s + 8·51-s + 24·53-s − 8·55-s − 12·57-s + 2·59-s + 2·61-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.894·5-s − 0.755·7-s − 1.20·11-s + 2.77·13-s + 1.03·15-s + 0.970·17-s − 1.37·19-s − 0.872·21-s − 4/5·25-s − 0.384·27-s + 0.742·29-s + 1.43·31-s − 1.39·33-s − 0.676·35-s + 0.657·37-s + 3.20·39-s + 0.624·41-s − 0.609·43-s − 1.16·47-s + 3/7·49-s + 1.12·51-s + 3.29·53-s − 1.07·55-s − 1.58·57-s + 0.260·59-s + 0.256·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3211264\)    =    \(2^{16} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(204.752\)
Root analytic conductor: \(3.78274\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3211264,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.188706359\)
\(L(\frac12)\) \(\approx\) \(4.188706359\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.3.ac_e
5$D_{4}$ \( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.5.ac_i
11$D_{4}$ \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.11.e_o
13$D_{4}$ \( 1 - 10 T + 48 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.13.ak_bw
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.17.ae_bm
19$D_{4}$ \( 1 + 6 T + 44 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.19.g_bs
23$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.23.a_bi
29$D_{4}$ \( 1 - 4 T + 50 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.29.ae_by
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.31.ai_da
37$D_{4}$ \( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.37.ae_co
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.41.ae_di
43$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.43.e_da
47$D_{4}$ \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.47.i_ck
53$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.53.ay_jq
59$D_{4}$ \( 1 - 2 T + 44 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.59.ac_bs
61$D_{4}$ \( 1 - 2 T - 24 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.61.ac_ay
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.67.aq_hq
71$D_{4}$ \( 1 - 8 T + 110 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.71.ai_eg
73$D_{4}$ \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.73.m_fe
79$D_{4}$ \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.79.i_ew
83$D_{4}$ \( 1 - 6 T + 100 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.83.ag_dw
89$D_{4}$ \( 1 - 4 T - 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.89.ae_ak
97$D_{4}$ \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.97.ae_fu
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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

−9.253040624711519209680139610122, −9.077653491582659354243822839238, −8.615530507070655399749285933414, −8.360032357897011971093428836927, −7.946863735446870377533255814020, −7.942631667902056021715410311813, −7.03322669551507082970663008197, −6.52661247881130734402729411472, −6.27520771628208186617024231429, −6.01290217464599460849970735609, −5.35585461296207624338836860423, −5.32452588655350428160513412822, −4.21857438777618662389274167994, −4.02280686643835579567134157531, −3.46869413088191558390132097313, −3.07206810596418610267219588385, −2.47151220982796584080128942268, −2.30081170083472991406564070794, −1.41069666208357436068907091684, −0.73758995759133091522638295855, 0.73758995759133091522638295855, 1.41069666208357436068907091684, 2.30081170083472991406564070794, 2.47151220982796584080128942268, 3.07206810596418610267219588385, 3.46869413088191558390132097313, 4.02280686643835579567134157531, 4.21857438777618662389274167994, 5.32452588655350428160513412822, 5.35585461296207624338836860423, 6.01290217464599460849970735609, 6.27520771628208186617024231429, 6.52661247881130734402729411472, 7.03322669551507082970663008197, 7.942631667902056021715410311813, 7.946863735446870377533255814020, 8.360032357897011971093428836927, 8.615530507070655399749285933414, 9.077653491582659354243822839238, 9.253040624711519209680139610122

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