Properties

Label 4-546e2-1.1-c1e2-0-59
Degree $4$
Conductor $298116$
Sign $1$
Analytic cond. $19.0081$
Root an. cond. $2.08802$
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  + 2-s + 3·3-s + 3·6-s + 5·7-s − 8-s + 6·9-s − 3·11-s + 7·13-s + 5·14-s − 16-s − 3·17-s + 6·18-s − 7·19-s + 15·21-s − 3·22-s + 12·23-s − 3·24-s − 2·25-s + 7·26-s + 9·27-s − 3·29-s − 8·31-s − 9·33-s − 3·34-s − 12·37-s − 7·38-s + 21·39-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.73·3-s + 1.22·6-s + 1.88·7-s − 0.353·8-s + 2·9-s − 0.904·11-s + 1.94·13-s + 1.33·14-s − 1/4·16-s − 0.727·17-s + 1.41·18-s − 1.60·19-s + 3.27·21-s − 0.639·22-s + 2.50·23-s − 0.612·24-s − 2/5·25-s + 1.37·26-s + 1.73·27-s − 0.557·29-s − 1.43·31-s − 1.56·33-s − 0.514·34-s − 1.97·37-s − 1.13·38-s + 3.36·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(298116\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(19.0081\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{546} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 298116,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.782905441\)
\(L(\frac12)\) \(\approx\) \(5.782905441\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3$C_2$ \( 1 - p T + p T^{2} \)
7$C_2$ \( 1 - 5 T + p T^{2} \)
13$C_2$ \( 1 - 7 T + p T^{2} \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 12 T + 71 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 9 T + 68 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 19 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 31 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 18 T + 167 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 9 T + 88 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 15 T + 164 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
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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

−10.84177431309224925357041016537, −10.82318833695876405541139087297, −10.30765734111193944929865928702, −9.507122661028961176946428581338, −8.954642786528935692383632500401, −8.583335330917070372761100024260, −8.399518132699041492276816647180, −8.358243979180356856401867568068, −7.29334407394395084745429439575, −7.20487540778183537262662955882, −6.66808747737944856848212619749, −5.67233096032637178991782036792, −5.35723241514589119346424560334, −4.85466274263137514455950931304, −4.07531622514595584239566222180, −3.98626857949496222011960525997, −3.28623842475651424666593322353, −2.55794422222255809473022854115, −1.94506129643303569782553429207, −1.40282037575030791985896656203, 1.40282037575030791985896656203, 1.94506129643303569782553429207, 2.55794422222255809473022854115, 3.28623842475651424666593322353, 3.98626857949496222011960525997, 4.07531622514595584239566222180, 4.85466274263137514455950931304, 5.35723241514589119346424560334, 5.67233096032637178991782036792, 6.66808747737944856848212619749, 7.20487540778183537262662955882, 7.29334407394395084745429439575, 8.358243979180356856401867568068, 8.399518132699041492276816647180, 8.583335330917070372761100024260, 8.954642786528935692383632500401, 9.507122661028961176946428581338, 10.30765734111193944929865928702, 10.82318833695876405541139087297, 10.84177431309224925357041016537

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