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Properties

Label 4-1568e2-1.1-c1e2-0-17
Degree $4$
Conductor $2458624$
Sign $1$
Analytic cond. $156.763$
Root an. cond. $3.53843$
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 + 3·9-s + 4·11-s + 12·13-s − 4·15-s + 4·17-s + 6·19-s − 4·23-s + 5·25-s + 10·27-s − 12·29-s − 4·31-s + 8·33-s + 6·37-s + 24·39-s + 8·41-s − 24·43-s − 6·45-s − 12·47-s + 8·51-s − 6·53-s − 8·55-s + 12·57-s + 6·59-s + 6·61-s − 24·65-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.894·5-s + 9-s + 1.20·11-s + 3.32·13-s − 1.03·15-s + 0.970·17-s + 1.37·19-s − 0.834·23-s + 25-s + 1.92·27-s − 2.22·29-s − 0.718·31-s + 1.39·33-s + 0.986·37-s + 3.84·39-s + 1.24·41-s − 3.65·43-s − 0.894·45-s − 1.75·47-s + 1.12·51-s − 0.824·53-s − 1.07·55-s + 1.58·57-s + 0.781·59-s + 0.768·61-s − 2.97·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2458624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2458624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2458624\)    =    \(2^{10} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(156.763\)
Root analytic conductor: \(3.53843\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2458624,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.596553608\)
\(L(\frac12)\) \(\approx\) \(4.596553608\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3$C_2^2$ \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.3.ac_b
5$C_2^2$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.5.c_ab
11$C_2^2$ \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.11.ae_f
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.13.am_ck
17$C_2^2$ \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.17.ae_ab
19$C_2^2$ \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.19.ag_r
23$C_2^2$ \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.23.e_ah
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.31.e_ap
37$C_2^2$ \( 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.37.ag_ab
41$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.41.ai_du
43$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.43.y_iw
47$C_2^2$ \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.47.m_dt
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.53.g_ar
59$C_2^2$ \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.59.ag_ax
61$C_2^2$ \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.61.ag_az
67$C_2^2$ \( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.67.am_cz
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.71.q_hy
73$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.73.a_acv
79$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.79.a_adb
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.83.m_hu
89$C_2^2$ \( 1 - 16 T + 167 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.89.aq_gl
97$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.97.y_na
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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

−9.584723174988188546993947756130, −9.103025715228052432040765742437, −8.791331992189608010518749960463, −8.339874052918427469718141485672, −8.166357092110904350859722254701, −7.86194302195830010849926626674, −7.31917379984719753624713250117, −6.79690802514786213284071376250, −6.53748010821381989165391206318, −6.06642648583743839640668251969, −5.54005617839634927804452474857, −5.11228148368691306244638780339, −4.30969781103385698378305403632, −3.83696140143315826391970569702, −3.73004876840804364129526399333, −3.18996788855871813113305332006, −3.08682253748482467970872824353, −1.65089572458930418899172354884, −1.57475572879696196979177965689, −0.886347392982711657919798619307, 0.886347392982711657919798619307, 1.57475572879696196979177965689, 1.65089572458930418899172354884, 3.08682253748482467970872824353, 3.18996788855871813113305332006, 3.73004876840804364129526399333, 3.83696140143315826391970569702, 4.30969781103385698378305403632, 5.11228148368691306244638780339, 5.54005617839634927804452474857, 6.06642648583743839640668251969, 6.53748010821381989165391206318, 6.79690802514786213284071376250, 7.31917379984719753624713250117, 7.86194302195830010849926626674, 8.166357092110904350859722254701, 8.339874052918427469718141485672, 8.791331992189608010518749960463, 9.103025715228052432040765742437, 9.584723174988188546993947756130

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