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Properties

Label 4-2883e2-1.1-c1e2-0-0
Degree $4$
Conductor $8311689$
Sign $1$
Analytic cond. $529.960$
Root an. cond. $4.79800$
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  − 2·3-s − 2·4-s − 4·5-s − 4·7-s + 3·9-s + 4·12-s − 2·13-s + 8·15-s + 8·17-s − 2·19-s + 8·20-s + 8·21-s − 4·23-s + 4·25-s − 4·27-s + 8·28-s − 4·29-s + 16·35-s − 6·36-s + 2·37-s + 4·39-s + 6·43-s − 12·45-s + 4·47-s + 6·49-s − 16·51-s + 4·52-s + ⋯
L(s)  = 1  − 1.15·3-s − 4-s − 1.78·5-s − 1.51·7-s + 9-s + 1.15·12-s − 0.554·13-s + 2.06·15-s + 1.94·17-s − 0.458·19-s + 1.78·20-s + 1.74·21-s − 0.834·23-s + 4/5·25-s − 0.769·27-s + 1.51·28-s − 0.742·29-s + 2.70·35-s − 36-s + 0.328·37-s + 0.640·39-s + 0.914·43-s − 1.78·45-s + 0.583·47-s + 6/7·49-s − 2.24·51-s + 0.554·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8311689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8311689 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8311689\)    =    \(3^{2} \cdot 31^{4}\)
Sign: $1$
Analytic conductor: \(529.960\)
Root analytic conductor: \(4.79800\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 8311689,\ (\ :1/2, 1/2),\ 1)\)

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 $\mathbf{F}_p$
bad3$C_1$ \( ( 1 + T )^{2} \)
31 \( 1 \)
good2$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.2.a_c
5$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.5.e_m
7$C_4$ \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.7.e_k
11$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.11.a_u
13$D_{4}$ \( 1 + 2 T + 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.13.c_t
17$D_{4}$ \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.17.ai_bq
19$D_{4}$ \( 1 + 2 T + 31 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.19.c_bf
23$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.23.e_s
29$D_{4}$ \( 1 + 4 T + 60 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.29.e_ci
37$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.37.ac_cx
41$C_2^2$ \( 1 + 64 T^{2} + p^{2} T^{4} \) 2.41.a_cm
43$D_{4}$ \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.43.ag_x
47$D_{4}$ \( 1 - 4 T + 96 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.47.ae_ds
53$C_2^2$ \( 1 + 104 T^{2} + p^{2} T^{4} \) 2.53.a_ea
59$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.59.a_bu
61$D_{4}$ \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.61.i_co
67$D_{4}$ \( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.67.m_du
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.71.am_gw
73$D_{4}$ \( 1 - 2 T + 139 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.73.ac_fj
79$D_{4}$ \( 1 - 20 T + 226 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.79.au_is
83$D_{4}$ \( 1 + 12 T + 200 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.83.m_hs
89$D_{4}$ \( 1 + 16 T + 144 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.89.q_fo
97$D_{4}$ \( 1 + 6 T + 131 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.97.g_fb
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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

−8.373166905996796479897185315831, −8.237137780706074720781002216008, −7.63930319983651167489926020606, −7.57298543722198551418129270716, −7.18890620322741594960278610711, −6.57680736321764174831536427892, −6.35727315716280202927335840733, −5.86668431895590555623422560858, −5.36429660867862093327325854009, −5.26270590275650783152496764163, −4.56146987489359038981656081379, −4.11461876509841742932086735053, −3.86427875009663268138519906397, −3.72118395883004514814902150366, −2.99205972758778751930206975744, −2.57341570869275308492125437365, −1.52897336538699596388933939554, −0.76100929863801819776434290851, 0, 0, 0.76100929863801819776434290851, 1.52897336538699596388933939554, 2.57341570869275308492125437365, 2.99205972758778751930206975744, 3.72118395883004514814902150366, 3.86427875009663268138519906397, 4.11461876509841742932086735053, 4.56146987489359038981656081379, 5.26270590275650783152496764163, 5.36429660867862093327325854009, 5.86668431895590555623422560858, 6.35727315716280202927335840733, 6.57680736321764174831536427892, 7.18890620322741594960278610711, 7.57298543722198551418129270716, 7.63930319983651167489926020606, 8.237137780706074720781002216008, 8.373166905996796479897185315831

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