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Properties

Label 4-630e2-1.1-c1e2-0-11
Degree $4$
Conductor $396900$
Sign $1$
Analytic cond. $25.3066$
Root an. cond. $2.24289$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 5-s − 7-s − 8-s − 3·9-s − 10-s − 3·11-s + 7·13-s − 14-s − 16-s − 6·17-s − 3·18-s + 4·19-s − 3·22-s + 6·23-s + 7·26-s − 6·29-s − 2·31-s − 6·34-s + 35-s + 4·37-s + 4·38-s + 40-s − 2·43-s + 3·45-s + 6·46-s + 9·47-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.447·5-s − 0.377·7-s − 0.353·8-s − 9-s − 0.316·10-s − 0.904·11-s + 1.94·13-s − 0.267·14-s − 1/4·16-s − 1.45·17-s − 0.707·18-s + 0.917·19-s − 0.639·22-s + 1.25·23-s + 1.37·26-s − 1.11·29-s − 0.359·31-s − 1.02·34-s + 0.169·35-s + 0.657·37-s + 0.648·38-s + 0.158·40-s − 0.304·43-s + 0.447·45-s + 0.884·46-s + 1.31·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(396900\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(25.3066\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 396900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.580728970\)
\(L(\frac12)\) \(\approx\) \(1.580728970\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3$C_2$ \( 1 + p T^{2} \)
5$C_2$ \( 1 + T + T^{2} \)
7$C_2$ \( 1 + T + T^{2} \)
good11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.11.d_ac
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.13.ah_bk
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.17.g_br
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.19.ae_bq
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_n
29$C_2^2$ \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.29.g_h
31$C_2^2$ \( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.31.c_abb
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.41.a_abp
43$C_2^2$ \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.43.c_abn
47$C_2^2$ \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.47.aj_bi
53$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.53.y_jq
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.59.a_ach
61$C_2$ \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.61.o_ff
67$C_2^2$ \( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.67.ak_bh
71$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.71.ag_fv
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.73.aw_kh
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.an_dm
83$C_2^2$ \( 1 - 15 T + 142 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.83.ap_fm
89$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \) 2.89.bk_ti
97$C_2^2$ \( 1 - 7 T - 48 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.97.ah_abw
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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

−11.02899682959039022957422339945, −10.85159304727180832664473029993, −9.852369508344136494152828533485, −9.278119897006643230124257513161, −9.207459433361387903284111137088, −8.656724238737901460993706924875, −8.044401932808785731129259804198, −7.967294812938959796112940887013, −7.19147285393687959779301365795, −6.69143538748531820733094173730, −6.10311326534115306126471650699, −5.97187231975871790783285480418, −5.16369681359052419291252148112, −4.97560946437625996538807041027, −4.23438658234391449612922307025, −3.57546554545646443228642307645, −3.30255262924032122707808302392, −2.72799219891489375905379303114, −1.86569059496937546483952543249, −0.61233565255700156135411541140, 0.61233565255700156135411541140, 1.86569059496937546483952543249, 2.72799219891489375905379303114, 3.30255262924032122707808302392, 3.57546554545646443228642307645, 4.23438658234391449612922307025, 4.97560946437625996538807041027, 5.16369681359052419291252148112, 5.97187231975871790783285480418, 6.10311326534115306126471650699, 6.69143538748531820733094173730, 7.19147285393687959779301365795, 7.967294812938959796112940887013, 8.044401932808785731129259804198, 8.656724238737901460993706924875, 9.207459433361387903284111137088, 9.278119897006643230124257513161, 9.852369508344136494152828533485, 10.85159304727180832664473029993, 11.02899682959039022957422339945

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