Properties

Label 4-1216e2-1.1-c1e2-0-12
Degree $4$
Conductor $1478656$
Sign $1$
Analytic cond. $94.2803$
Root an. cond. $3.11605$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 2·5-s − 4·7-s + 3·9-s + 6·11-s + 2·13-s + 6·15-s + 6·17-s − 7·19-s − 12·21-s + 4·23-s + 5·25-s − 6·29-s − 16·31-s + 18·33-s − 8·35-s + 16·37-s + 6·39-s + 3·41-s − 8·43-s + 6·45-s − 10·47-s − 2·49-s + 18·51-s + 2·53-s + 12·55-s − 21·57-s + ⋯
L(s)  = 1  + 1.73·3-s + 0.894·5-s − 1.51·7-s + 9-s + 1.80·11-s + 0.554·13-s + 1.54·15-s + 1.45·17-s − 1.60·19-s − 2.61·21-s + 0.834·23-s + 25-s − 1.11·29-s − 2.87·31-s + 3.13·33-s − 1.35·35-s + 2.63·37-s + 0.960·39-s + 0.468·41-s − 1.21·43-s + 0.894·45-s − 1.45·47-s − 2/7·49-s + 2.52·51-s + 0.274·53-s + 1.61·55-s − 2.78·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1478656\)    =    \(2^{12} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(94.2803\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1478656,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.640071234\)
\(L(\frac12)\) \(\approx\) \(4.640071234\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_2$ \( 1 + 7 T + p T^{2} \)
good3$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T^{2} ) \)
5$C_2^2$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 10 T + 53 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \)
67$C_2^2$ \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 2 T - 67 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 3 T - 88 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
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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

−10.00431386422429153414374754127, −9.380223639214603638906155702403, −9.252287473325924229936402473775, −8.704909694898883205662825513555, −8.670621924200620095138294850563, −7.85957994609888695610652195154, −7.67501456563732144815196020146, −6.98010381012137288010553702983, −6.54428525324287580171982652725, −6.32503752002096581382086677644, −5.93937603347040367790963498606, −5.36545148723838319832003762488, −4.73229448925364767081950572092, −3.93056353121384108972772128180, −3.54496741903122418953904651114, −3.41075162547430933125434319580, −2.89369548827754809882945450080, −2.06183208017385057947220202493, −1.83239589263981842342506649680, −0.840316365765853048516606605503, 0.840316365765853048516606605503, 1.83239589263981842342506649680, 2.06183208017385057947220202493, 2.89369548827754809882945450080, 3.41075162547430933125434319580, 3.54496741903122418953904651114, 3.93056353121384108972772128180, 4.73229448925364767081950572092, 5.36545148723838319832003762488, 5.93937603347040367790963498606, 6.32503752002096581382086677644, 6.54428525324287580171982652725, 6.98010381012137288010553702983, 7.67501456563732144815196020146, 7.85957994609888695610652195154, 8.670621924200620095138294850563, 8.704909694898883205662825513555, 9.252287473325924229936402473775, 9.380223639214603638906155702403, 10.00431386422429153414374754127

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