Properties

Label 4-3240e2-1.1-c1e2-0-38
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 $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 4·11-s − 4·19-s − 25-s − 14·29-s − 12·31-s − 10·41-s + 13·49-s − 8·55-s − 24·59-s − 14·61-s + 20·71-s − 8·79-s − 30·89-s − 8·95-s − 36·101-s + 18·109-s − 10·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s − 28·145-s + 149-s + 151-s − 24·155-s + ⋯
L(s)  = 1  + 0.894·5-s − 1.20·11-s − 0.917·19-s − 1/5·25-s − 2.59·29-s − 2.15·31-s − 1.56·41-s + 13/7·49-s − 1.07·55-s − 3.12·59-s − 1.79·61-s + 2.37·71-s − 0.900·79-s − 3.17·89-s − 0.820·95-s − 3.58·101-s + 1.72·109-s − 0.909·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.32·145-s + 0.0819·149-s + 0.0813·151-s − 1.92·155-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: \(2\)
Selberg data: \((4,\ 10497600,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad2 \( 1 \)
3 \( 1 \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
good7$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 109 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 141 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
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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

−8.333850492704480769851561105597, −8.215570316728486187696690032595, −7.53778154627210261660722986118, −7.53700775575874945712663074933, −6.89839777946427120402596687372, −6.74134447359235885255018317771, −6.00668718229200828867710256337, −5.74457616380981758760380316808, −5.41776516276226923318697331497, −5.33511539359141794047150278471, −4.55669047760570695677804463018, −4.22087584125780946286703516023, −3.71305143930328964300848470200, −3.26969219563027547973866839063, −2.75145904941817198324972458428, −2.20892907972174812096932151549, −1.76109338713353075897667098336, −1.51653118959106214770588220927, 0, 0, 1.51653118959106214770588220927, 1.76109338713353075897667098336, 2.20892907972174812096932151549, 2.75145904941817198324972458428, 3.26969219563027547973866839063, 3.71305143930328964300848470200, 4.22087584125780946286703516023, 4.55669047760570695677804463018, 5.33511539359141794047150278471, 5.41776516276226923318697331497, 5.74457616380981758760380316808, 6.00668718229200828867710256337, 6.74134447359235885255018317771, 6.89839777946427120402596687372, 7.53700775575874945712663074933, 7.53778154627210261660722986118, 8.215570316728486187696690032595, 8.333850492704480769851561105597

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