Properties

Label 4-6300e2-1.1-c1e2-0-6
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  − 6·11-s − 4·19-s + 18·29-s + 16·31-s + 12·41-s − 49-s − 20·61-s − 30·71-s + 14·79-s − 24·89-s − 36·101-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 + ⋯
L(s)  = 1  − 1.80·11-s − 0.917·19-s + 3.34·29-s + 2.87·31-s + 1.87·41-s − 1/7·49-s − 2.56·61-s − 3.56·71-s + 1.57·79-s − 2.54·89-s − 3.58·101-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 + ⋯

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

\(L(1)\) \(\approx\) \(1.378897669\)
\(L(\frac12)\) \(\approx\) \(1.378897669\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 49 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 109 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−8.412412552174874456933094093239, −7.946110573982799486091420010926, −7.59673282857246887469174827491, −7.22707766623033468348974339257, −6.72315927607361845491847168706, −6.29586370172444276615365349420, −6.28223934468593390261912801886, −5.78204595803649042373411526002, −5.36370588995301521812360886280, −4.81062052791549533876991006955, −4.64390732814656640974551887554, −4.35263356116546961295477839035, −4.04248090932884763695847172434, −3.09151127709324076511899581190, −2.79229716152413812994803029694, −2.74170557376387366199668566427, −2.35935998883737133930097634312, −1.39936001262246305795840570720, −1.10947405025221599273854521377, −0.32101073849467955773732352010, 0.32101073849467955773732352010, 1.10947405025221599273854521377, 1.39936001262246305795840570720, 2.35935998883737133930097634312, 2.74170557376387366199668566427, 2.79229716152413812994803029694, 3.09151127709324076511899581190, 4.04248090932884763695847172434, 4.35263356116546961295477839035, 4.64390732814656640974551887554, 4.81062052791549533876991006955, 5.36370588995301521812360886280, 5.78204595803649042373411526002, 6.28223934468593390261912801886, 6.29586370172444276615365349420, 6.72315927607361845491847168706, 7.22707766623033468348974339257, 7.59673282857246887469174827491, 7.946110573982799486091420010926, 8.412412552174874456933094093239

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