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Properties

Label 4-7440e2-1.1-c1e2-0-6
Degree $4$
Conductor $55353600$
Sign $1$
Analytic cond. $3529.39$
Root an. cond. $7.70770$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 2·5-s − 6·7-s + 3·9-s − 4·11-s + 6·13-s − 4·15-s − 4·19-s + 12·21-s + 3·25-s − 4·27-s + 2·29-s + 2·31-s + 8·33-s − 12·35-s + 2·37-s − 12·39-s + 6·45-s + 8·47-s + 16·49-s − 8·55-s + 8·57-s + 18·59-s − 8·61-s − 18·63-s + 12·65-s − 2·67-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.894·5-s − 2.26·7-s + 9-s − 1.20·11-s + 1.66·13-s − 1.03·15-s − 0.917·19-s + 2.61·21-s + 3/5·25-s − 0.769·27-s + 0.371·29-s + 0.359·31-s + 1.39·33-s − 2.02·35-s + 0.328·37-s − 1.92·39-s + 0.894·45-s + 1.16·47-s + 16/7·49-s − 1.07·55-s + 1.05·57-s + 2.34·59-s − 1.02·61-s − 2.26·63-s + 1.48·65-s − 0.244·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55353600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55353600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(55353600\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(3529.39\)
Root analytic conductor: \(7.70770\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 55353600,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
3$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
31$C_1$ \( ( 1 - T )^{2} \)
good7$D_{4}$ \( 1 + 6 T + 20 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.7.g_u
11$D_{4}$ \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.11.e_o
13$D_{4}$ \( 1 - 6 T + 32 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.13.ag_bg
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.19.e_be
23$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.23.a_bi
29$D_{4}$ \( 1 - 2 T + 56 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.29.ac_ce
37$D_{4}$ \( 1 - 2 T - 2 p T^{3} + p^{2} T^{4} \) 2.37.ac_a
41$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.41.a_cs
43$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.43.a_di
47$D_{4}$ \( 1 - 8 T + 98 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.47.ai_du
53$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.53.a_ec
59$D_{4}$ \( 1 - 18 T + 172 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.59.as_gq
61$D_{4}$ \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.61.i_ew
67$D_{4}$ \( 1 + 2 T + 60 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.67.c_ci
71$D_{4}$ \( 1 + 10 T + 92 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.71.k_do
73$D_{4}$ \( 1 - 6 T + 152 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.73.ag_fw
79$D_{4}$ \( 1 + 20 T + 210 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.79.u_ic
83$C_2^2$ \( 1 + 154 T^{2} + p^{2} T^{4} \) 2.83.a_fy
89$D_{4}$ \( 1 + 10 T + 200 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.89.k_hs
97$D_{4}$ \( 1 + 20 T + 246 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.97.u_jm
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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

−7.46035978401060098112633059110, −7.24907027235218332739606175987, −6.79603219668824691600971120053, −6.68772724795432565616018094182, −6.16357367221948373812278333063, −5.95686212145390998742946196953, −5.75532816814066068401505846838, −5.63067142113783302128806190033, −4.86834445092147678921427087062, −4.67764496955062078738501296619, −3.95168722776987332456468938256, −3.94086482050235609201523536202, −3.14859723283124513809546001205, −3.09985544304040067302173100582, −2.33054228433801212715461922406, −2.27305098981560058310421218870, −1.17823033746030481941427751423, −1.09892093923063247444360354762, 0, 0, 1.09892093923063247444360354762, 1.17823033746030481941427751423, 2.27305098981560058310421218870, 2.33054228433801212715461922406, 3.09985544304040067302173100582, 3.14859723283124513809546001205, 3.94086482050235609201523536202, 3.95168722776987332456468938256, 4.67764496955062078738501296619, 4.86834445092147678921427087062, 5.63067142113783302128806190033, 5.75532816814066068401505846838, 5.95686212145390998742946196953, 6.16357367221948373812278333063, 6.68772724795432565616018094182, 6.79603219668824691600971120053, 7.24907027235218332739606175987, 7.46035978401060098112633059110

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