Properties

Label 4-6300e2-1.1-c1e2-0-7
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

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 6·11-s + 8·19-s + 6·29-s − 20·31-s − 49-s + 24·59-s − 8·61-s − 18·71-s − 34·79-s − 12·101-s + 14·109-s + 5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 1.80·11-s + 1.83·19-s + 1.11·29-s − 3.59·31-s − 1/7·49-s + 3.12·59-s − 1.02·61-s − 2.13·71-s − 3.82·79-s − 1.19·101-s + 1.34·109-s + 5/11·121-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 + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-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

\(L(1)\) \(\approx\) \(2.682814204\)
\(L(\frac12)\) \(\approx\) \(2.682814204\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 17 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.322096589156685691598145824695, −7.68519459455951485668326698699, −7.46427488600430883842200283114, −7.11560338521717918413242425983, −6.92111606218338561244311284497, −6.56603745670835960160324422235, −6.01781934009883028351677982200, −5.64973004598675539874019358222, −5.49030598156504651563292690032, −5.10196232464633745481516783849, −4.51058905661888656209865764273, −4.08762263087621523526595477272, −3.93222993704373413354317890318, −3.34950228022828918976509199505, −3.15503970886952016409268439764, −2.60224800943285292542151671419, −1.95178073243425631827428459389, −1.35017779617208586007762528395, −1.33645845069426978825235813149, −0.42757675158975702221547456367, 0.42757675158975702221547456367, 1.33645845069426978825235813149, 1.35017779617208586007762528395, 1.95178073243425631827428459389, 2.60224800943285292542151671419, 3.15503970886952016409268439764, 3.34950228022828918976509199505, 3.93222993704373413354317890318, 4.08762263087621523526595477272, 4.51058905661888656209865764273, 5.10196232464633745481516783849, 5.49030598156504651563292690032, 5.64973004598675539874019358222, 6.01781934009883028351677982200, 6.56603745670835960160324422235, 6.92111606218338561244311284497, 7.11560338521717918413242425983, 7.46427488600430883842200283114, 7.68519459455951485668326698699, 8.322096589156685691598145824695

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