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Properties

Label 4-2240e2-1.1-c1e2-0-19
Degree $4$
Conductor $5017600$
Sign $1$
Analytic cond. $319.926$
Root an. cond. $4.22924$
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  − 3-s + 2·5-s − 2·7-s − 9-s − 7·11-s + 3·13-s − 2·15-s − 17-s − 8·19-s + 2·21-s + 3·25-s − 5·29-s + 2·31-s + 7·33-s − 4·35-s − 10·37-s − 3·39-s − 2·45-s + 9·47-s + 3·49-s + 51-s + 6·53-s − 14·55-s + 8·57-s − 12·59-s + 2·63-s + 6·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s − 0.755·7-s − 1/3·9-s − 2.11·11-s + 0.832·13-s − 0.516·15-s − 0.242·17-s − 1.83·19-s + 0.436·21-s + 3/5·25-s − 0.928·29-s + 0.359·31-s + 1.21·33-s − 0.676·35-s − 1.64·37-s − 0.480·39-s − 0.298·45-s + 1.31·47-s + 3/7·49-s + 0.140·51-s + 0.824·53-s − 1.88·55-s + 1.05·57-s − 1.56·59-s + 0.251·63-s + 0.744·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5017600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5017600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5017600\)    =    \(2^{12} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(319.926\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 5017600,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) 2.3.b_c
11$D_{4}$ \( 1 + 7 T + 30 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.11.h_be
13$D_{4}$ \( 1 - 3 T + 24 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.13.ad_y
17$D_{4}$ \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) 2.17.b_ae
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$D_{4}$ \( 1 + 5 T + 60 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.29.f_ci
31$D_{4}$ \( 1 - 2 T + 46 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.31.ac_bu
37$D_{4}$ \( 1 + 10 T + 82 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.37.k_de
41$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.41.a_o
43$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.43.a_di
47$D_{4}$ \( 1 - 9 T + 110 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.47.aj_eg
53$D_{4}$ \( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.53.ag_du
59$D_{4}$ \( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.59.m_di
61$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \) 2.61.a_cc
67$D_{4}$ \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.67.m_dy
71$D_{4}$ \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.71.am_eg
73$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \) 2.73.a_da
79$D_{4}$ \( 1 + 7 T + 166 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.79.h_gk
83$D_{4}$ \( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.83.e_dy
89$D_{4}$ \( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.89.q_gs
97$D_{4}$ \( 1 + T + 156 T^{2} + p T^{3} + p^{2} T^{4} \) 2.97.b_ga
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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

−8.760975126704104472798126933685, −8.691889320129556478339366763614, −8.057961367228640787276719687245, −7.72000079586027864267527122650, −7.20480465582430065115977063450, −6.87113633648011159920068264641, −6.25731794299334339414704817840, −6.16246587607197873764207343281, −5.57741415942539649970952661523, −5.54599048835834242428033548529, −4.93738826760294885342796553823, −4.54155038574519082095010134071, −3.87293143840550308815480320051, −3.52856452610335365195324901719, −2.66009562584759757029644016706, −2.62320767673832926271838277920, −1.99198778744404568029366434298, −1.30115390466568493480286475090, 0, 0, 1.30115390466568493480286475090, 1.99198778744404568029366434298, 2.62320767673832926271838277920, 2.66009562584759757029644016706, 3.52856452610335365195324901719, 3.87293143840550308815480320051, 4.54155038574519082095010134071, 4.93738826760294885342796553823, 5.54599048835834242428033548529, 5.57741415942539649970952661523, 6.16246587607197873764207343281, 6.25731794299334339414704817840, 6.87113633648011159920068264641, 7.20480465582430065115977063450, 7.72000079586027864267527122650, 8.057961367228640787276719687245, 8.691889320129556478339366763614, 8.760975126704104472798126933685

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