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Properties

Label 4-645e2-1.1-c1e2-0-4
Degree $4$
Conductor $416025$
Sign $1$
Analytic cond. $26.5261$
Root an. cond. $2.26943$
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  − 2·3-s − 2·4-s − 2·5-s + 3·9-s − 2·11-s + 4·12-s − 2·13-s + 4·15-s + 6·17-s − 12·19-s + 4·20-s + 2·23-s + 3·25-s − 4·27-s − 10·31-s + 4·33-s − 6·36-s − 8·37-s + 4·39-s − 2·41-s − 2·43-s + 4·44-s − 6·45-s − 4·47-s − 12·49-s − 12·51-s + 4·52-s + ⋯
L(s)  = 1  − 1.15·3-s − 4-s − 0.894·5-s + 9-s − 0.603·11-s + 1.15·12-s − 0.554·13-s + 1.03·15-s + 1.45·17-s − 2.75·19-s + 0.894·20-s + 0.417·23-s + 3/5·25-s − 0.769·27-s − 1.79·31-s + 0.696·33-s − 36-s − 1.31·37-s + 0.640·39-s − 0.312·41-s − 0.304·43-s + 0.603·44-s − 0.894·45-s − 0.583·47-s − 1.71·49-s − 1.68·51-s + 0.554·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(416025\)    =    \(3^{2} \cdot 5^{2} \cdot 43^{2}\)
Sign: $1$
Analytic conductor: \(26.5261\)
Root analytic conductor: \(2.26943\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 416025,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
43$C_1$ \( ( 1 + T )^{2} \)
good2$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.2.a_c
7$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \) 2.7.a_m
11$D_{4}$ \( 1 + 2 T + 21 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.11.c_v
13$D_{4}$ \( 1 + 2 T + 9 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.13.c_j
17$D_{4}$ \( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.17.ag_bj
19$C_2^2$ \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.19.m_cu
23$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.23.ac_bv
29$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.29.a_by
31$D_{4}$ \( 1 + 10 T + 79 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.31.k_db
37$D_{4}$ \( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.37.i_de
41$D_{4}$ \( 1 + 2 T + 81 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.41.c_dd
47$D_{4}$ \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.47.e_dm
53$D_{4}$ \( 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.53.ag_df
59$D_{4}$ \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.59.i_ew
61$D_{4}$ \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.61.i_co
67$D_{4}$ \( 1 + 2 T + 117 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.67.c_en
71$C_2^2$ \( 1 + 124 T^{2} + p^{2} T^{4} \) 2.71.a_eu
73$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.73.a_fq
79$D_{4}$ \( 1 - 4 T + 154 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.79.ae_fy
83$D_{4}$ \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.83.g_az
89$D_{4}$ \( 1 - 16 T + 224 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.89.aq_iq
97$D_{4}$ \( 1 + 10 T + 121 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.97.k_er
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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

−10.35125905352681040284523681625, −10.22959359202811319606860551463, −9.321427457682005521780802979389, −9.214172307365881797231462124464, −8.494201877992222164795775575360, −8.260696901557407710627771940853, −7.64272717185928922189673254627, −7.33127065196266707683181802294, −6.71022550151055970429019747872, −6.32281330911682488468235756354, −5.72414407781646793399326289735, −5.12778551605474033940845589497, −4.83162809834509413773597466339, −4.48873361601401164359794802789, −3.70214419165706644951983782538, −3.48474942075808157329737067321, −2.36113121337548442900681845434, −1.49376989590777575546609911799, 0, 0, 1.49376989590777575546609911799, 2.36113121337548442900681845434, 3.48474942075808157329737067321, 3.70214419165706644951983782538, 4.48873361601401164359794802789, 4.83162809834509413773597466339, 5.12778551605474033940845589497, 5.72414407781646793399326289735, 6.32281330911682488468235756354, 6.71022550151055970429019747872, 7.33127065196266707683181802294, 7.64272717185928922189673254627, 8.260696901557407710627771940853, 8.494201877992222164795775575360, 9.214172307365881797231462124464, 9.321427457682005521780802979389, 10.22959359202811319606860551463, 10.35125905352681040284523681625

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