Properties

Label 4-6300e2-1.1-c1e2-0-20
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 $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·7-s − 20·31-s + 22·37-s + 2·43-s + 3·49-s − 16·61-s − 6·67-s − 20·73-s − 22·79-s + 4·97-s − 16·103-s + 6·109-s − 15·121-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  − 0.755·7-s − 3.59·31-s + 3.61·37-s + 0.304·43-s + 3/7·49-s − 2.04·61-s − 0.733·67-s − 2.34·73-s − 2.47·79-s + 0.406·97-s − 1.57·103-s + 0.574·109-s − 1.36·121-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: \(2\)
Selberg data: \((4,\ 39690000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 + 135 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 138 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 66 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−7.57394593727947946680112997706, −7.57121392498381895646330943976, −7.13000219741371933222216237726, −7.07712273589617818696312533218, −6.24124185082791768897128872502, −6.03618916637823544309374218199, −5.81616487820076713533972090578, −5.62823896137632088010913486177, −4.80120337544866269247187749279, −4.68806156840476633748816827653, −4.19558364897568658077687378417, −3.76155901299700067163746741018, −3.49256157607282103092925543093, −2.93718423611550946149078680490, −2.57663510062400056104961919527, −2.21664253462857908695625960050, −1.37191111072859135387070609048, −1.24911715806644303642649261474, 0, 0, 1.24911715806644303642649261474, 1.37191111072859135387070609048, 2.21664253462857908695625960050, 2.57663510062400056104961919527, 2.93718423611550946149078680490, 3.49256157607282103092925543093, 3.76155901299700067163746741018, 4.19558364897568658077687378417, 4.68806156840476633748816827653, 4.80120337544866269247187749279, 5.62823896137632088010913486177, 5.81616487820076713533972090578, 6.03618916637823544309374218199, 6.24124185082791768897128872502, 7.07712273589617818696312533218, 7.13000219741371933222216237726, 7.57121392498381895646330943976, 7.57394593727947946680112997706

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