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Properties

Label 4-1776e2-1.1-c1e2-0-0
Degree $4$
Conductor $3154176$
Sign $1$
Analytic cond. $201.112$
Root an. cond. $3.76582$
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  + 3-s − 5-s + 2·7-s − 4·11-s − 6·13-s − 15-s − 3·17-s − 2·19-s + 2·21-s − 16·23-s + 5·25-s − 27-s + 2·29-s − 4·31-s − 4·33-s − 2·35-s − 11·37-s − 6·39-s + 9·41-s − 20·43-s − 4·47-s + 7·49-s − 3·51-s + 2·53-s + 4·55-s − 2·57-s + 12·59-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 0.755·7-s − 1.20·11-s − 1.66·13-s − 0.258·15-s − 0.727·17-s − 0.458·19-s + 0.436·21-s − 3.33·23-s + 25-s − 0.192·27-s + 0.371·29-s − 0.718·31-s − 0.696·33-s − 0.338·35-s − 1.80·37-s − 0.960·39-s + 1.40·41-s − 3.04·43-s − 0.583·47-s + 49-s − 0.420·51-s + 0.274·53-s + 0.539·55-s − 0.264·57-s + 1.56·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3154176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3154176 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3154176\)    =    \(2^{8} \cdot 3^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(201.112\)
Root analytic conductor: \(3.76582\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3154176,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1201625518\)
\(L(\frac12)\) \(\approx\) \(0.1201625518\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( 1 - T + T^{2} \)
37$C_2$ \( 1 + 11 T + p T^{2} \)
good5$C_2^2$ \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) 2.5.b_ae
7$C_2^2$ \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.7.ac_ad
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.11.e_ba
13$C_2^2$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.13.g_x
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.17.d_ai
19$C_2^2$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.19.c_ap
23$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.23.q_eg
29$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.29.ac_ch
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.31.e_co
41$C_2^2$ \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.41.aj_bo
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.43.u_he
47$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.47.e_du
53$C_2^2$ \( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.53.ac_abx
59$C_2^2$ \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.59.am_dh
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.61.b_aci
67$C_2^2$ \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.67.ac_acl
71$C_2^2$ \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.71.ag_abj
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.73.e_fu
79$C_2^2$ \( 1 + 10 T + 21 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.79.k_v
83$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.83.a_adf
89$C_2^2$ \( 1 - T - 88 T^{2} - p T^{3} + p^{2} T^{4} \) 2.89.ab_adk
97$C_2$ \( ( 1 + 17 T + p T^{2} )^{2} \) 2.97.bi_sp
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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

−9.585984941451943900585119147895, −8.819342440378634549982473047204, −8.706688683881323865981340602352, −8.211505345120875757081600668477, −7.88779701932594154971806374484, −7.80030516675932434636503806294, −7.16817347986889534429021156387, −6.75386645616522036401396811429, −6.52366500702814598380076618972, −5.56565862751748986366169002006, −5.47699713641819690411385778398, −4.98198740643296251961005963769, −4.54239972967074432242139475703, −4.01502911730563238981725264782, −3.76605848323980246189780561087, −2.90630400922222438516478703343, −2.54316594726131580774649942335, −2.04042857643770901830625056349, −1.66839199175126427980426171702, −0.11379031289299978359594767081, 0.11379031289299978359594767081, 1.66839199175126427980426171702, 2.04042857643770901830625056349, 2.54316594726131580774649942335, 2.90630400922222438516478703343, 3.76605848323980246189780561087, 4.01502911730563238981725264782, 4.54239972967074432242139475703, 4.98198740643296251961005963769, 5.47699713641819690411385778398, 5.56565862751748986366169002006, 6.52366500702814598380076618972, 6.75386645616522036401396811429, 7.16817347986889534429021156387, 7.80030516675932434636503806294, 7.88779701932594154971806374484, 8.211505345120875757081600668477, 8.706688683881323865981340602352, 8.819342440378634549982473047204, 9.585984941451943900585119147895

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