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Properties

Label 4-5054e2-1.1-c1e2-0-6
Degree $4$
Conductor $25542916$
Sign $1$
Analytic cond. $1628.63$
Root an. cond. $6.35266$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3-s + 3·4-s − 5-s − 2·6-s − 2·7-s − 4·8-s − 9-s + 2·10-s − 9·11-s + 3·12-s − 5·13-s + 4·14-s − 15-s + 5·16-s + 4·17-s + 2·18-s − 3·20-s − 2·21-s + 18·22-s + 3·23-s − 4·24-s − 5·25-s + 10·26-s − 6·28-s − 4·29-s + 2·30-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s + 3/2·4-s − 0.447·5-s − 0.816·6-s − 0.755·7-s − 1.41·8-s − 1/3·9-s + 0.632·10-s − 2.71·11-s + 0.866·12-s − 1.38·13-s + 1.06·14-s − 0.258·15-s + 5/4·16-s + 0.970·17-s + 0.471·18-s − 0.670·20-s − 0.436·21-s + 3.83·22-s + 0.625·23-s − 0.816·24-s − 25-s + 1.96·26-s − 1.13·28-s − 0.742·29-s + 0.365·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25542916 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25542916 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(25542916\)    =    \(2^{2} \cdot 7^{2} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(1628.63\)
Root analytic conductor: \(6.35266\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 25542916,\ (\ :1/2, 1/2),\ 1)\)

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 $\mathbf{F}_p$
bad2$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
19 \( 1 \)
good3$D_{4}$ \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) 2.3.ab_c
5$D_{4}$ \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) 2.5.b_g
11$C_2^2$ \( 1 + 9 T + 38 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.11.j_bm
13$D_{4}$ \( 1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.13.f_bc
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.17.ae_bm
23$D_{4}$ \( 1 - 3 T + 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.23.ad_bs
29$D_{4}$ \( 1 + 4 T - 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.29.e_ag
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.31.e_co
37$D_{4}$ \( 1 - 6 T + 66 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.37.ag_co
41$D_{4}$ \( 1 + 3 T + 46 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.41.d_bu
43$D_{4}$ \( 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.43.g_da
47$D_{4}$ \( 1 + 6 T + 86 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.47.g_di
53$C_4$ \( 1 - 10 T + 114 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.53.ak_ek
59$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.59.aw_jf
61$D_{4}$ \( 1 + 25 T + 274 T^{2} + 25 p T^{3} + p^{2} T^{4} \) 2.61.z_ko
67$D_{4}$ \( 1 - 5 T + 102 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.67.af_dy
71$D_{4}$ \( 1 + 9 T + 56 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.71.j_ce
73$D_{4}$ \( 1 - 5 T + 148 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.73.af_fs
79$D_{4}$ \( 1 - 12 T + 126 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.79.am_ew
83$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.83.c_gl
89$D_{4}$ \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.89.ag_bi
97$D_{4}$ \( 1 - 11 T + 220 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.97.al_im
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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

−7.938699140937639407942253480574, −7.76995769402685508933090579984, −7.50101783970925971195529880459, −7.36514741694795920594298522843, −6.74755899223569457547927976336, −6.52145831223007962725186688641, −5.77397714662177227051718741054, −5.61206238635669801138200175016, −5.15613519116512244095218322243, −5.01231298280523420693828797195, −4.25081088536293641574505687853, −3.70510269013130893257637203443, −3.12769526624377122873826365303, −3.05755649708403679259600780303, −2.44100410852304892721369829543, −2.33773556504755878410887672981, −1.71183351456946573959201295640, −0.796313127913351791602583661283, 0, 0, 0.796313127913351791602583661283, 1.71183351456946573959201295640, 2.33773556504755878410887672981, 2.44100410852304892721369829543, 3.05755649708403679259600780303, 3.12769526624377122873826365303, 3.70510269013130893257637203443, 4.25081088536293641574505687853, 5.01231298280523420693828797195, 5.15613519116512244095218322243, 5.61206238635669801138200175016, 5.77397714662177227051718741054, 6.52145831223007962725186688641, 6.74755899223569457547927976336, 7.36514741694795920594298522843, 7.50101783970925971195529880459, 7.76995769402685508933090579984, 7.938699140937639407942253480574

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