Properties

Label 4-3240e2-1.1-c1e2-0-25
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 7-s − 11-s − 13-s − 2·17-s + 2·19-s + 5·23-s + 3·25-s − 29-s + 5·31-s + 2·35-s + 12·37-s + 12·41-s + 2·43-s − 13·47-s − 5·49-s + 3·53-s − 2·55-s + 10·59-s + 14·61-s − 2·65-s + 10·67-s − 3·71-s + 18·73-s − 77-s + 6·79-s − 20·83-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.377·7-s − 0.301·11-s − 0.277·13-s − 0.485·17-s + 0.458·19-s + 1.04·23-s + 3/5·25-s − 0.185·29-s + 0.898·31-s + 0.338·35-s + 1.97·37-s + 1.87·41-s + 0.304·43-s − 1.89·47-s − 5/7·49-s + 0.412·53-s − 0.269·55-s + 1.30·59-s + 1.79·61-s − 0.248·65-s + 1.22·67-s − 0.356·71-s + 2.10·73-s − 0.113·77-s + 0.675·79-s − 2.19·83-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

\(L(1)\) \(\approx\) \(4.156602791\)
\(L(\frac12)\) \(\approx\) \(4.156602791\)
\(L(\frac{3}{2})\) 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$D_{4}$ \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + T + 14 T^{2} + p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 - 5 T + 44 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + T + 50 T^{2} + p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 2 T + 54 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 13 T + 128 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 3 T + 34 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 - 14 T + 138 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 10 T + 126 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 3 T + 136 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 18 T + 194 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 6 T + 134 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 + 3 T + 172 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 2 T + 162 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
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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

−8.698511993935674131969216115964, −8.519795717253944912663599767676, −8.127776229435482206257163639049, −7.79284459499562321459638171151, −7.29615789752344524872698395955, −6.93233445152459762775167582449, −6.65707135671478672315207342489, −6.15435169817612484984230038921, −5.80597376598047505137023348460, −5.46860093726808454451869375991, −4.94930839222967786169274911446, −4.73846595381922484920980904226, −4.27180001033612376583202349250, −3.78095438132272975838659211095, −3.08439032990115222115262600494, −2.82634096317416370562432484461, −2.13626437367416932207927540651, −2.05625676713037649228336489776, −0.987532938810142313325817271311, −0.77600406385429099299892574913, 0.77600406385429099299892574913, 0.987532938810142313325817271311, 2.05625676713037649228336489776, 2.13626437367416932207927540651, 2.82634096317416370562432484461, 3.08439032990115222115262600494, 3.78095438132272975838659211095, 4.27180001033612376583202349250, 4.73846595381922484920980904226, 4.94930839222967786169274911446, 5.46860093726808454451869375991, 5.80597376598047505137023348460, 6.15435169817612484984230038921, 6.65707135671478672315207342489, 6.93233445152459762775167582449, 7.29615789752344524872698395955, 7.79284459499562321459638171151, 8.127776229435482206257163639049, 8.519795717253944912663599767676, 8.698511993935674131969216115964

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