Properties

Label 4-6300e2-1.1-c1e2-0-12
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·7-s − 4·13-s + 4·19-s + 4·31-s + 2·37-s − 10·43-s + 3·49-s + 16·61-s + 14·67-s − 16·73-s + 10·79-s + 8·91-s + 20·97-s − 4·103-s − 2·109-s + 5·121-s + 127-s + 131-s − 8·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 14·169-s + ⋯
L(s)  = 1  − 0.755·7-s − 1.10·13-s + 0.917·19-s + 0.718·31-s + 0.328·37-s − 1.52·43-s + 3/7·49-s + 2.04·61-s + 1.71·67-s − 1.87·73-s + 1.12·79-s + 0.838·91-s + 2.03·97-s − 0.394·103-s − 0.191·109-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s − 0.693·133-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 − 1.07·169-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

\(L(1)\) \(\approx\) \(2.524579939\)
\(L(\frac12)\) \(\approx\) \(2.524579939\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
good11$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 31 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 + 115 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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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

−8.079827647138800709655667568088, −8.009453558555650146854530908810, −7.29052118447827194730304903333, −7.24864177144473226148728766918, −6.83274747774986761647498516345, −6.53653295086721472746579570588, −6.11635716448353117977695011511, −5.73657458636930658568946142645, −5.23405696528098221058433277587, −5.15933903721764150552906732447, −4.56936313411491332175208358602, −4.31082379899903763448916238423, −3.70105857299602315751269251191, −3.43519736911905592307639602491, −2.82698501253899719782525062103, −2.80070195812088469099053808016, −1.96244211369667752975547251027, −1.80252283893987559868457194371, −0.75048225872808911157737227977, −0.56108504410263755734081397261, 0.56108504410263755734081397261, 0.75048225872808911157737227977, 1.80252283893987559868457194371, 1.96244211369667752975547251027, 2.80070195812088469099053808016, 2.82698501253899719782525062103, 3.43519736911905592307639602491, 3.70105857299602315751269251191, 4.31082379899903763448916238423, 4.56936313411491332175208358602, 5.15933903721764150552906732447, 5.23405696528098221058433277587, 5.73657458636930658568946142645, 6.11635716448353117977695011511, 6.53653295086721472746579570588, 6.83274747774986761647498516345, 7.24864177144473226148728766918, 7.29052118447827194730304903333, 8.009453558555650146854530908810, 8.079827647138800709655667568088

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