Properties

Label 4-3240e2-1.1-c1e2-0-30
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  + 5-s + 4·7-s + 4·11-s + 2·13-s − 4·17-s + 8·19-s + 4·23-s − 2·29-s + 8·31-s + 4·35-s + 12·37-s − 6·41-s + 8·43-s + 4·47-s + 7·49-s − 12·53-s + 4·55-s − 4·59-s + 2·61-s + 2·65-s − 8·67-s − 12·73-s + 16·77-s − 16·83-s − 4·85-s + 12·89-s + 8·91-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.51·7-s + 1.20·11-s + 0.554·13-s − 0.970·17-s + 1.83·19-s + 0.834·23-s − 0.371·29-s + 1.43·31-s + 0.676·35-s + 1.97·37-s − 0.937·41-s + 1.21·43-s + 0.583·47-s + 49-s − 1.64·53-s + 0.539·55-s − 0.520·59-s + 0.256·61-s + 0.248·65-s − 0.977·67-s − 1.40·73-s + 1.82·77-s − 1.75·83-s − 0.433·85-s + 1.27·89-s + 0.838·91-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

\(L(1)\) \(\approx\) \(5.434640946\)
\(L(\frac12)\) \(\approx\) \(5.434640946\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
good7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2^2$ \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 4 T - 31 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 16 T + 173 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 19 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
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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

−8.775738221971129928713017847475, −8.620925990908311200375000500775, −8.055497835935780895990818239749, −7.70580819100260057630002073703, −7.39564614325142566789176250416, −7.10097507023005863153467861656, −6.45354592331089491291581107900, −6.25473549548713818342062133531, −5.79283828259738763106229961254, −5.52935531655825512705437523890, −4.77255057291672011978712871596, −4.66002107074911457044767329640, −4.39401867612762282632330780686, −3.79866559309019276402115184434, −3.14499468958761546637218678166, −2.92894620361717558564372310535, −2.16028693128308461113817397367, −1.71580801263595136428293652893, −1.15445220581255516115920080329, −0.855327904736472794737847602444, 0.855327904736472794737847602444, 1.15445220581255516115920080329, 1.71580801263595136428293652893, 2.16028693128308461113817397367, 2.92894620361717558564372310535, 3.14499468958761546637218678166, 3.79866559309019276402115184434, 4.39401867612762282632330780686, 4.66002107074911457044767329640, 4.77255057291672011978712871596, 5.52935531655825512705437523890, 5.79283828259738763106229961254, 6.25473549548713818342062133531, 6.45354592331089491291581107900, 7.10097507023005863153467861656, 7.39564614325142566789176250416, 7.70580819100260057630002073703, 8.055497835935780895990818239749, 8.620925990908311200375000500775, 8.775738221971129928713017847475

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