Properties

Label 4-6300e2-1.1-c1e2-0-11
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  + 2·11-s + 4·19-s + 2·29-s + 12·31-s − 49-s − 12·59-s + 16·61-s − 14·71-s − 2·79-s − 28·89-s + 24·101-s + 14·109-s − 19·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  + 0.603·11-s + 0.917·19-s + 0.371·29-s + 2.15·31-s − 1/7·49-s − 1.56·59-s + 2.04·61-s − 1.66·71-s − 0.225·79-s − 2.96·89-s + 2.38·101-s + 1.34·109-s − 1.72·121-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 + 1.69·169-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

\(L(1)\) \(\approx\) \(3.416629824\)
\(L(\frac12)\) \(\approx\) \(3.416629824\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 125 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
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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

−8.269202740732184609784950497556, −7.85402300740634683499642464402, −7.53009940962773471629759904841, −7.11240221074108072589695786607, −6.68807402918247619527077030385, −6.67943831844853787327134434182, −5.96915945365031783969578428443, −5.80251415198006960203902236078, −5.49488355413082216615212915254, −4.86066726143237848397469325973, −4.47662606159424866129878131770, −4.45668484684864758960679609174, −3.80436166487709745462738680674, −3.36056250607840505835607877714, −2.92521723457689171434467502489, −2.73759464070514296062738072543, −2.00306222542157523385055120376, −1.57435841845552287555689404896, −0.993147858956797702808122568249, −0.54702362048965279976334571577, 0.54702362048965279976334571577, 0.993147858956797702808122568249, 1.57435841845552287555689404896, 2.00306222542157523385055120376, 2.73759464070514296062738072543, 2.92521723457689171434467502489, 3.36056250607840505835607877714, 3.80436166487709745462738680674, 4.45668484684864758960679609174, 4.47662606159424866129878131770, 4.86066726143237848397469325973, 5.49488355413082216615212915254, 5.80251415198006960203902236078, 5.96915945365031783969578428443, 6.67943831844853787327134434182, 6.68807402918247619527077030385, 7.11240221074108072589695786607, 7.53009940962773471629759904841, 7.85402300740634683499642464402, 8.269202740732184609784950497556

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