Properties

Label 4-6300e2-1.1-c1e2-0-18
Degree $4$
Conductor $39690000$
Sign $1$
Analytic cond. $2530.66$
Root an. cond. $7.09265$
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  + 8·11-s + 12·19-s + 4·31-s + 4·41-s − 49-s − 8·59-s − 4·61-s + 16·71-s + 16·79-s − 20·89-s − 36·101-s + 36·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  + 2.41·11-s + 2.75·19-s + 0.718·31-s + 0.624·41-s − 1/7·49-s − 1.04·59-s − 0.512·61-s + 1.89·71-s + 1.80·79-s − 2.11·89-s − 3.58·101-s + 3.44·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39690000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39690000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(39690000\)    =    \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(2530.66\)
Root analytic conductor: \(7.09265\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{6300} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 39690000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.227912494\)
\(L(\frac12)\) \(\approx\) \(5.227912494\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 + T^{2} \)
good11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 178 T^{2} + p^{2} T^{4} \)
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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

−8.029108341433804675935561384812, −8.021279672209943821296496575223, −7.36217632658600584537799457631, −7.19873817642127874573907845842, −6.73897017268950462542234465879, −6.61013293141771302146961983835, −5.97913968347512812799097645355, −5.90739157938273370534202987837, −5.32345802526354629006121808286, −5.06611641492879727186260982347, −4.47395119402303431507444268712, −4.29429827690586609545079306104, −3.63328223819437140553514499652, −3.60172147668940021473344557986, −2.95338828867454867458710314230, −2.77480262112386879330596720038, −1.80914884569144606538662082038, −1.62610686909669391694051832092, −0.926987279583274509990761737795, −0.75185787420926044424284759151, 0.75185787420926044424284759151, 0.926987279583274509990761737795, 1.62610686909669391694051832092, 1.80914884569144606538662082038, 2.77480262112386879330596720038, 2.95338828867454867458710314230, 3.60172147668940021473344557986, 3.63328223819437140553514499652, 4.29429827690586609545079306104, 4.47395119402303431507444268712, 5.06611641492879727186260982347, 5.32345802526354629006121808286, 5.90739157938273370534202987837, 5.97913968347512812799097645355, 6.61013293141771302146961983835, 6.73897017268950462542234465879, 7.19873817642127874573907845842, 7.36217632658600584537799457631, 8.021279672209943821296496575223, 8.029108341433804675935561384812

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