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Properties

Label 4-630e2-1.1-c1e2-0-70
Degree $4$
Conductor $396900$
Sign $1$
Analytic cond. $25.3066$
Root an. cond. $2.24289$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3·3-s + 3·4-s − 5-s + 6·6-s − 7-s − 4·8-s + 6·9-s + 2·10-s − 9·12-s − 2·13-s + 2·14-s + 3·15-s + 5·16-s − 12·18-s − 2·19-s − 3·20-s + 3·21-s + 12·24-s + 4·26-s − 9·27-s − 3·28-s − 9·29-s − 6·30-s − 8·31-s − 6·32-s + 35-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.73·3-s + 3/2·4-s − 0.447·5-s + 2.44·6-s − 0.377·7-s − 1.41·8-s + 2·9-s + 0.632·10-s − 2.59·12-s − 0.554·13-s + 0.534·14-s + 0.774·15-s + 5/4·16-s − 2.82·18-s − 0.458·19-s − 0.670·20-s + 0.654·21-s + 2.44·24-s + 0.784·26-s − 1.73·27-s − 0.566·28-s − 1.67·29-s − 1.09·30-s − 1.43·31-s − 1.06·32-s + 0.169·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(396900\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(25.3066\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 396900,\ (\ :1/2, 1/2),\ 1)\)

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$C_2$ \( 1 + p T + p T^{2} \)
5$C_2$ \( 1 + T + T^{2} \)
7$C_2$ \( 1 + T + p T^{2} \)
good11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.11.a_al
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.13.c_aj
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.17.a_ar
19$C_2^2$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.19.c_ap
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.23.a_ax
29$C_2^2$ \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.29.j_ca
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$C_2^2$ \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.37.c_abh
41$C_2^2$ \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.41.ad_abg
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.43.f_as
47$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.47.s_gt
53$C_2^2$ \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.53.m_dn
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.59.am_fy
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.61.u_io
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.67.i_fu
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.71.m_gw
73$C_2^2$ \( 1 + 8 T - 9 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.73.i_aj
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.79.aq_io
83$C_2^2$ \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.83.j_ac
89$C_2^2$ \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.89.g_acb
97$C_2^2$ \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.97.ak_d
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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

−10.31588875935745795208413781780, −10.08965789392874766315017603931, −9.441059882757633026448659941786, −9.361469303294492760968238914149, −8.729650244708553949583419633140, −8.079672459695154435841726705262, −7.68246697990669215026946088794, −7.31789886586117032859422630741, −6.78787901492335549861798827823, −6.54145691846282398273810138875, −5.77085980274682732295768288850, −5.74803525457187469512994772055, −4.81762924018274443763444759378, −4.55081947928213342217778257302, −3.57012788919793055736804976877, −3.13670966998521605001242466226, −1.91664370796084748685425521482, −1.49592432160490729146114110591, 0, 0, 1.49592432160490729146114110591, 1.91664370796084748685425521482, 3.13670966998521605001242466226, 3.57012788919793055736804976877, 4.55081947928213342217778257302, 4.81762924018274443763444759378, 5.74803525457187469512994772055, 5.77085980274682732295768288850, 6.54145691846282398273810138875, 6.78787901492335549861798827823, 7.31789886586117032859422630741, 7.68246697990669215026946088794, 8.079672459695154435841726705262, 8.729650244708553949583419633140, 9.361469303294492760968238914149, 9.441059882757633026448659941786, 10.08965789392874766315017603931, 10.31588875935745795208413781780

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