Properties

Label 4-2646e2-1.1-c1e2-0-13
Degree $4$
Conductor $7001316$
Sign $1$
Analytic cond. $446.409$
Root an. cond. $4.59656$
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  + 2·2-s + 3·4-s + 3·5-s + 4·8-s + 6·10-s − 6·11-s + 2·13-s + 5·16-s − 6·17-s − 7·19-s + 9·20-s − 12·22-s + 3·23-s + 5·25-s + 4·26-s + 6·29-s − 4·31-s + 6·32-s − 12·34-s − 2·37-s − 14·38-s + 12·40-s − 2·43-s − 18·44-s + 6·46-s + 10·50-s + 6·52-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 1.34·5-s + 1.41·8-s + 1.89·10-s − 1.80·11-s + 0.554·13-s + 5/4·16-s − 1.45·17-s − 1.60·19-s + 2.01·20-s − 2.55·22-s + 0.625·23-s + 25-s + 0.784·26-s + 1.11·29-s − 0.718·31-s + 1.06·32-s − 2.05·34-s − 0.328·37-s − 2.27·38-s + 1.89·40-s − 0.304·43-s − 2.71·44-s + 0.884·46-s + 1.41·50-s + 0.832·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7001316\)    =    \(2^{2} \cdot 3^{6} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(446.409\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7001316,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.518876706\)
\(L(\frac12)\) \(\approx\) \(6.518876706\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
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

−9.281555630135559416055333377262, −8.582802745365980176714549917598, −8.123721029730607122251098581030, −8.123063428493459659180928151609, −7.24043210112474355261780469862, −6.87844093293021943315136714684, −6.76896785249808213671159573099, −6.14115125480109365377394483201, −5.98342890635902755421339175316, −5.47657645947162229343466478161, −5.21566168260829610499324672835, −4.72150674143783035808953228403, −4.45053261262822048545312171887, −3.99476402641683646171195354192, −3.31419653659577975557109894294, −2.80735051271798484923763990792, −2.54395127821462212286025034497, −1.91200897076389879440871610987, −1.81132020339380924657786265295, −0.59350255717347315057394342410, 0.59350255717347315057394342410, 1.81132020339380924657786265295, 1.91200897076389879440871610987, 2.54395127821462212286025034497, 2.80735051271798484923763990792, 3.31419653659577975557109894294, 3.99476402641683646171195354192, 4.45053261262822048545312171887, 4.72150674143783035808953228403, 5.21566168260829610499324672835, 5.47657645947162229343466478161, 5.98342890635902755421339175316, 6.14115125480109365377394483201, 6.76896785249808213671159573099, 6.87844093293021943315136714684, 7.24043210112474355261780469862, 8.123063428493459659180928151609, 8.123721029730607122251098581030, 8.582802745365980176714549917598, 9.281555630135559416055333377262

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