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Properties

Label 4-2106e2-1.1-c1e2-0-35
Degree $4$
Conductor $4435236$
Sign $1$
Analytic cond. $282.794$
Root an. cond. $4.10079$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s + 2·5-s + 2·7-s − 4·8-s − 4·10-s − 8·11-s − 2·13-s − 4·14-s + 5·16-s + 6·20-s + 16·22-s + 2·23-s − 4·25-s + 4·26-s + 6·28-s − 16·29-s + 8·31-s − 6·32-s + 4·35-s − 12·37-s − 8·40-s − 6·41-s − 4·43-s − 24·44-s − 4·46-s − 8·47-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 0.894·5-s + 0.755·7-s − 1.41·8-s − 1.26·10-s − 2.41·11-s − 0.554·13-s − 1.06·14-s + 5/4·16-s + 1.34·20-s + 3.41·22-s + 0.417·23-s − 4/5·25-s + 0.784·26-s + 1.13·28-s − 2.97·29-s + 1.43·31-s − 1.06·32-s + 0.676·35-s − 1.97·37-s − 1.26·40-s − 0.937·41-s − 0.609·43-s − 3.61·44-s − 0.589·46-s − 1.16·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4435236 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4435236 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4435236\)    =    \(2^{2} \cdot 3^{8} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(282.794\)
Root analytic conductor: \(4.10079\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 4435236,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 + T )^{2} \)
3 \( 1 \)
13$C_1$ \( ( 1 + T )^{2} \)
good5$D_{4}$ \( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.5.ac_i
7$D_{4}$ \( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.7.ac_d
11$D_{4}$ \( 1 + 8 T + 35 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.11.i_bj
17$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.17.a_ao
19$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \) 2.19.a_bj
23$D_{4}$ \( 1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.23.ac_u
29$D_{4}$ \( 1 + 16 T + 119 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.29.q_ep
31$D_{4}$ \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.31.ai_co
37$D_{4}$ \( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.37.m_du
41$D_{4}$ \( 1 + 6 T + 88 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.41.g_dk
43$D_{4}$ \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.43.e_bq
47$D_{4}$ \( 1 + 8 T + 98 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.47.i_du
53$C_2^2$ \( 1 + 31 T^{2} + p^{2} T^{4} \) 2.53.a_bf
59$D_{4}$ \( 1 + 4 T - 25 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.59.e_az
61$D_{4}$ \( 1 + 4 T + 99 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.61.e_dv
67$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.67.e_be
71$D_{4}$ \( 1 + 14 T + 179 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.71.o_gx
73$D_{4}$ \( 1 + 20 T + 234 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.73.u_ja
79$D_{4}$ \( 1 - 22 T + 252 T^{2} - 22 p T^{3} + p^{2} T^{4} \) 2.79.aw_js
83$D_{4}$ \( 1 + 12 T + 127 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.83.m_ex
89$D_{4}$ \( 1 - 18 T + 232 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.89.as_iy
97$D_{4}$ \( 1 - 22 T + 312 T^{2} - 22 p T^{3} + p^{2} T^{4} \) 2.97.aw_ma
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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

−8.907087079188679987349114491578, −8.551114141832779612209200415234, −8.128566454951051357982948207637, −7.71176653963758680456671370099, −7.52762529710846723322839369210, −7.30175441238869628610376847070, −6.67390066077669515326965706375, −6.10442559899069989064364781896, −5.85847139206074798344123280612, −5.31840368647455923100676110416, −4.95256159029416531923328418827, −4.84688452923613685087150192246, −3.75744449828790083603113475907, −3.32195737050018164203386438444, −2.65052789659500072464411331977, −2.35737155687156421438091072128, −1.65009900268749605715328001033, −1.59742641325301888831450611010, 0, 0, 1.59742641325301888831450611010, 1.65009900268749605715328001033, 2.35737155687156421438091072128, 2.65052789659500072464411331977, 3.32195737050018164203386438444, 3.75744449828790083603113475907, 4.84688452923613685087150192246, 4.95256159029416531923328418827, 5.31840368647455923100676110416, 5.85847139206074798344123280612, 6.10442559899069989064364781896, 6.67390066077669515326965706375, 7.30175441238869628610376847070, 7.52762529710846723322839369210, 7.71176653963758680456671370099, 8.128566454951051357982948207637, 8.551114141832779612209200415234, 8.907087079188679987349114491578

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