Properties

Label 4-3240e2-1.1-c1e2-0-28
Degree $4$
Conductor $10497600$
Sign $1$
Analytic cond. $669.336$
Root an. cond. $5.08640$
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  + 2·5-s + 2·7-s + 4·17-s − 4·19-s + 10·23-s + 3·25-s − 6·29-s + 12·31-s + 4·35-s − 12·37-s + 10·41-s + 4·43-s + 14·47-s − 5·49-s + 4·53-s + 12·59-s − 6·61-s − 2·67-s + 8·73-s − 4·79-s + 6·83-s + 8·85-s + 14·89-s − 8·95-s + 4·97-s + 4·101-s + 20·103-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.755·7-s + 0.970·17-s − 0.917·19-s + 2.08·23-s + 3/5·25-s − 1.11·29-s + 2.15·31-s + 0.676·35-s − 1.97·37-s + 1.56·41-s + 0.609·43-s + 2.04·47-s − 5/7·49-s + 0.549·53-s + 1.56·59-s − 0.768·61-s − 0.244·67-s + 0.936·73-s − 0.450·79-s + 0.658·83-s + 0.867·85-s + 1.48·89-s − 0.820·95-s + 0.406·97-s + 0.398·101-s + 1.97·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10497600\)    =    \(2^{6} \cdot 3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(669.336\)
Root analytic conductor: \(5.08640\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3240} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10497600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.792890515\)
\(L(\frac12)\) \(\approx\) \(4.792890515\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
3 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
good7$D_{4}$ \( 1 - 2 T + 9 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 10 T + 65 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 12 T + 74 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 - 10 T + 83 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T - 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 14 T + 137 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 12 T + 130 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 6 T + 169 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 14 T + 131 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−8.810094473244902225863958671273, −8.708854249634062873437283848740, −7.923685810313280565397389556111, −7.86872472112905429783454606288, −7.29561105727588080836615868616, −7.05642717890901162506008104396, −6.46727990629738871001075173803, −6.32629228830983142066046800463, −5.59886534851593283043038125112, −5.56554023586073573580900961129, −4.99341645588020936344215002534, −4.76715643472403162551356374357, −4.18775743972261135793208126128, −3.81827485177166097005396021959, −3.08777468354378049008845670357, −2.88796835386999305826815644374, −2.07683890516739483340949947699, −2.00506994364080859634970460265, −1.01004956177176863900695628427, −0.841177527231864577721067200852, 0.841177527231864577721067200852, 1.01004956177176863900695628427, 2.00506994364080859634970460265, 2.07683890516739483340949947699, 2.88796835386999305826815644374, 3.08777468354378049008845670357, 3.81827485177166097005396021959, 4.18775743972261135793208126128, 4.76715643472403162551356374357, 4.99341645588020936344215002534, 5.56554023586073573580900961129, 5.59886534851593283043038125112, 6.32629228830983142066046800463, 6.46727990629738871001075173803, 7.05642717890901162506008104396, 7.29561105727588080836615868616, 7.86872472112905429783454606288, 7.923685810313280565397389556111, 8.708854249634062873437283848740, 8.810094473244902225863958671273

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