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Properties

Label 4-7605e2-1.1-c1e2-0-9
Degree $4$
Conductor $57836025$
Sign $1$
Analytic cond. $3687.67$
Root an. cond. $7.79270$
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  − 2·2-s + 2·4-s − 2·5-s − 2·7-s − 4·8-s + 4·10-s + 4·14-s + 8·16-s + 10·17-s + 4·19-s − 4·20-s + 8·23-s + 3·25-s − 4·28-s + 2·29-s − 4·31-s − 8·32-s − 20·34-s + 4·35-s − 8·37-s − 8·38-s + 8·40-s − 14·41-s − 4·43-s − 16·46-s + 10·47-s + 49-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 0.894·5-s − 0.755·7-s − 1.41·8-s + 1.26·10-s + 1.06·14-s + 2·16-s + 2.42·17-s + 0.917·19-s − 0.894·20-s + 1.66·23-s + 3/5·25-s − 0.755·28-s + 0.371·29-s − 0.718·31-s − 1.41·32-s − 3.42·34-s + 0.676·35-s − 1.31·37-s − 1.29·38-s + 1.26·40-s − 2.18·41-s − 0.609·43-s − 2.35·46-s + 1.45·47-s + 1/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57836025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57836025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(57836025\)    =    \(3^{4} \cdot 5^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(3687.67\)
Root analytic conductor: \(7.79270\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 57836025,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
13 \( 1 \)
good2$C_2^2$ \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) 2.2.c_c
7$D_{4}$ \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_d
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.11.a_k
17$D_{4}$ \( 1 - 10 T + 56 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.17.ak_ce
19$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.19.ae_be
23$D_{4}$ \( 1 - 8 T + 50 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.23.ai_by
29$D_{4}$ \( 1 - 2 T + 56 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.29.ac_ce
31$D_{4}$ \( 1 + 4 T + 39 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.31.e_bn
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.37.i_dm
41$D_{4}$ \( 1 + 14 T + 128 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.41.o_ey
43$D_{4}$ \( 1 + 4 T + 87 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.43.e_dj
47$D_{4}$ \( 1 - 10 T + 92 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.47.ak_do
53$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.53.a_cg
59$D_{4}$ \( 1 + 18 T + 196 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.59.s_ho
61$D_{4}$ \( 1 - 2 T + 111 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.61.ac_eh
67$D_{4}$ \( 1 + 18 T + 203 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.67.s_hv
71$D_{4}$ \( 1 + 22 T + 260 T^{2} + 22 p T^{3} + p^{2} T^{4} \) 2.71.w_ka
73$D_{4}$ \( 1 - 10 T + 63 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.73.ak_cl
79$D_{4}$ \( 1 + 10 T + 135 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.79.k_ff
83$D_{4}$ \( 1 + 12 T + 190 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.83.m_hi
89$D_{4}$ \( 1 + 6 T + 184 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.89.g_hc
97$D_{4}$ \( 1 - 26 T + 351 T^{2} - 26 p T^{3} + p^{2} T^{4} \) 2.97.aba_nn
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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

−7.54148415053992095658560073741, −7.48681871367425135466848299423, −7.30693820048572745049057502828, −6.79049618465863224590740014512, −6.51283018523610979781662079205, −5.93071805067082377411300814757, −5.70536095288057817996588194593, −5.42790696920699000623971358107, −4.77696576748326439747505972507, −4.76864451751183392411411144937, −3.73013629301501020589259344855, −3.61119893830739079532843018727, −3.23086089839306556243710949580, −2.87643957144702388216065273453, −2.79488583044377622843072655319, −1.62527163086152550526846431693, −1.31313546234294107033929749413, −0.966497683811036615818317136827, 0, 0, 0.966497683811036615818317136827, 1.31313546234294107033929749413, 1.62527163086152550526846431693, 2.79488583044377622843072655319, 2.87643957144702388216065273453, 3.23086089839306556243710949580, 3.61119893830739079532843018727, 3.73013629301501020589259344855, 4.76864451751183392411411144937, 4.77696576748326439747505972507, 5.42790696920699000623971358107, 5.70536095288057817996588194593, 5.93071805067082377411300814757, 6.51283018523610979781662079205, 6.79049618465863224590740014512, 7.30693820048572745049057502828, 7.48681871367425135466848299423, 7.54148415053992095658560073741

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