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Properties

Label 4-558e2-1.1-c1e2-0-5
Degree $4$
Conductor $311364$
Sign $1$
Analytic cond. $19.8528$
Root an. cond. $2.11084$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s − 3·5-s + 7-s + 4·8-s − 6·10-s + 3·11-s − 5·13-s + 2·14-s + 5·16-s + 3·17-s + 7·19-s − 9·20-s + 6·22-s + 5·25-s − 10·26-s + 3·28-s − 12·29-s − 4·31-s + 6·32-s + 6·34-s − 3·35-s + 7·37-s + 14·38-s − 12·40-s + 3·41-s − 5·43-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 1.34·5-s + 0.377·7-s + 1.41·8-s − 1.89·10-s + 0.904·11-s − 1.38·13-s + 0.534·14-s + 5/4·16-s + 0.727·17-s + 1.60·19-s − 2.01·20-s + 1.27·22-s + 25-s − 1.96·26-s + 0.566·28-s − 2.22·29-s − 0.718·31-s + 1.06·32-s + 1.02·34-s − 0.507·35-s + 1.15·37-s + 2.27·38-s − 1.89·40-s + 0.468·41-s − 0.762·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 311364 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311364 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(311364\)    =    \(2^{2} \cdot 3^{4} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(19.8528\)
Root analytic conductor: \(2.11084\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 311364,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.805439512\)
\(L(\frac12)\) \(\approx\) \(3.805439512\)
\(L(\frac{3}{2})\) 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 \( 1 \)
31$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2^2$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.5.d_e
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.ab_ag
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.11.ad_ac
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.13.f_m
17$C_2^2$ \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.17.ad_ai
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.19.ah_be
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
37$C_2^2$ \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.37.ah_m
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$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.47.ay_je
53$C_2^2$ \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.53.aj_bc
59$C_2^2$ \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.59.d_aby
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.61.u_io
67$C_2^2$ \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.67.an_dy
71$C_2^2$ \( 1 + 3 T - 62 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.71.d_ack
73$C_2^2$ \( 1 - 13 T + 96 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.73.an_ds
79$C_2^2$ \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) 2.79.ab_ada
83$C_2^2$ \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.83.j_ac
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.97.ae_hq
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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

−11.22275943386260477331696097932, −10.79993670702986353973431177711, −10.30030455167212410108549701495, −9.645577726413583026226755782301, −9.268378053049533752810937914799, −8.865049762419181923665004600083, −7.998721917769142684388514085320, −7.53534120707006903524393337716, −7.30508308588942493336678610469, −7.27294586570844070070191643806, −6.34870032885874448028992261736, −5.60778260663906808050098259181, −5.49015616129293459542646032424, −4.87603865224511495515451097049, −4.12113625375092048794348631041, −4.01838039073801222850824626673, −3.40897137906598837640320817197, −2.78245278083099687735775899975, −2.03662367753396185886817406214, −0.949734282541615877198605666525, 0.949734282541615877198605666525, 2.03662367753396185886817406214, 2.78245278083099687735775899975, 3.40897137906598837640320817197, 4.01838039073801222850824626673, 4.12113625375092048794348631041, 4.87603865224511495515451097049, 5.49015616129293459542646032424, 5.60778260663906808050098259181, 6.34870032885874448028992261736, 7.27294586570844070070191643806, 7.30508308588942493336678610469, 7.53534120707006903524393337716, 7.998721917769142684388514085320, 8.865049762419181923665004600083, 9.268378053049533752810937914799, 9.645577726413583026226755782301, 10.30030455167212410108549701495, 10.79993670702986353973431177711, 11.22275943386260477331696097932

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