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Properties

Label 4-5712e2-1.1-c1e2-0-2
Degree $4$
Conductor $32626944$
Sign $1$
Analytic cond. $2080.32$
Root an. cond. $6.75355$
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  − 2·3-s − 3·5-s + 2·7-s + 3·9-s − 7·11-s + 13-s + 6·15-s − 2·17-s + 6·19-s − 4·21-s + 6·23-s + 25-s − 4·27-s − 16·31-s + 14·33-s − 6·35-s + 7·37-s − 2·39-s + 14·41-s − 15·43-s − 9·45-s − 16·47-s + 3·49-s + 4·51-s + 3·53-s + 21·55-s − 12·57-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.34·5-s + 0.755·7-s + 9-s − 2.11·11-s + 0.277·13-s + 1.54·15-s − 0.485·17-s + 1.37·19-s − 0.872·21-s + 1.25·23-s + 1/5·25-s − 0.769·27-s − 2.87·31-s + 2.43·33-s − 1.01·35-s + 1.15·37-s − 0.320·39-s + 2.18·41-s − 2.28·43-s − 1.34·45-s − 2.33·47-s + 3/7·49-s + 0.560·51-s + 0.412·53-s + 2.83·55-s − 1.58·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32626944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32626944 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(32626944\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(2080.32\)
Root analytic conductor: \(6.75355\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 32626944,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.085471496\)
\(L(\frac12)\) \(\approx\) \(1.085471496\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
17$C_1$ \( ( 1 + T )^{2} \)
good5$C_2^2$ \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.5.d_i
11$D_{4}$ \( 1 + 7 T + 30 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.11.h_be
13$C_2^2$ \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) 2.13.ab_am
19$D_{4}$ \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.19.ag_be
23$D_{4}$ \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_bm
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.29.a_ak
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.31.q_ew
37$D_{4}$ \( 1 - 7 T + 48 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.37.ah_bw
41$D_{4}$ \( 1 - 14 T + 114 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.41.ao_ek
43$D_{4}$ \( 1 + 15 T + 138 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.43.p_fi
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.47.q_gc
53$D_{4}$ \( 1 - 3 T + 104 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.53.ad_ea
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.59.i_fe
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.61.e_ew
67$D_{4}$ \( 1 - 3 T + 98 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.67.ad_du
71$D_{4}$ \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.71.am_eg
73$D_{4}$ \( 1 - 15 T + 164 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.73.ap_gi
79$D_{4}$ \( 1 - 7 T + 166 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.79.ah_gk
83$D_{4}$ \( 1 + 13 T + 170 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.83.n_go
89$D_{4}$ \( 1 - 19 T + 264 T^{2} - 19 p T^{3} + p^{2} T^{4} \) 2.89.at_ke
97$D_{4}$ \( 1 - 27 T + 372 T^{2} - 27 p T^{3} + p^{2} T^{4} \) 2.97.abb_oi
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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

−7.989225787399187307779335174638, −7.82421552109261245101092341784, −7.59597315329272320494037174514, −7.45360478561353335826796273051, −6.96222231638771360017059525648, −6.58288354508234442825331501380, −6.02740286337534784466313001178, −5.76866091507498555124765446134, −5.24231420979671537039604108297, −5.06998544905286549631578770957, −4.76246842215950926495776152189, −4.56710282705782533223290804686, −3.79413252909244216867603455568, −3.48622765927943495814725213061, −3.24829794108260367532279464655, −2.58340458711967822035476814784, −1.96577994631501865148168382198, −1.62780839715320124975554819180, −0.58789663510534964853950696107, −0.52732602809611688722583711070, 0.52732602809611688722583711070, 0.58789663510534964853950696107, 1.62780839715320124975554819180, 1.96577994631501865148168382198, 2.58340458711967822035476814784, 3.24829794108260367532279464655, 3.48622765927943495814725213061, 3.79413252909244216867603455568, 4.56710282705782533223290804686, 4.76246842215950926495776152189, 5.06998544905286549631578770957, 5.24231420979671537039604108297, 5.76866091507498555124765446134, 6.02740286337534784466313001178, 6.58288354508234442825331501380, 6.96222231638771360017059525648, 7.45360478561353335826796273051, 7.59597315329272320494037174514, 7.82421552109261245101092341784, 7.989225787399187307779335174638

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