Properties

Label 4-546e2-1.1-c1e2-0-74
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 $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 6-s − 7-s + 8-s − 3·11-s − 7·13-s + 14-s − 16-s − 3·17-s − 5·19-s + 21-s + 3·22-s − 6·23-s − 24-s − 10·25-s + 7·26-s + 27-s + 3·29-s − 8·31-s + 3·33-s + 3·34-s + 4·37-s + 5·38-s + 7·39-s + 3·41-s − 42-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.408·6-s − 0.377·7-s + 0.353·8-s − 0.904·11-s − 1.94·13-s + 0.267·14-s − 1/4·16-s − 0.727·17-s − 1.14·19-s + 0.218·21-s + 0.639·22-s − 1.25·23-s − 0.204·24-s − 2·25-s + 1.37·26-s + 0.192·27-s + 0.557·29-s − 1.43·31-s + 0.522·33-s + 0.514·34-s + 0.657·37-s + 0.811·38-s + 1.12·39-s + 0.468·41-s − 0.154·42-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: \(2\)
Selberg data: \((4,\ 298116,\ (\ :1/2, 1/2),\ 1)\)

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$C_2$ \( 1 + T + T^{2} \)
3$C_2$ \( 1 + T + T^{2} \)
7$C_2$ \( 1 + T + T^{2} \)
13$C_2$ \( 1 + 7 T + p T^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
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$C_2^2$ \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
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$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
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$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 9 T - 8 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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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

−10.37202673827620200804606497013, −10.23495964714052628902466770313, −9.651121492418243418918015037962, −9.453642667954677657666806461203, −8.897255014354849941293596911901, −8.317257046262649002426373369172, −7.83873337026315190808385447530, −7.55815905293563325856043959559, −7.09243760743157993969694933263, −6.50536205682837573880549311290, −5.87080297661846121160247021845, −5.66492534122420757550052588777, −4.92195010975224217259434907813, −4.37589542230758019747744243792, −4.05536305569503541777473301077, −3.00578576155251164584655802355, −2.30054975701406865933678141656, −1.87798864913775327366154204737, 0, 0, 1.87798864913775327366154204737, 2.30054975701406865933678141656, 3.00578576155251164584655802355, 4.05536305569503541777473301077, 4.37589542230758019747744243792, 4.92195010975224217259434907813, 5.66492534122420757550052588777, 5.87080297661846121160247021845, 6.50536205682837573880549311290, 7.09243760743157993969694933263, 7.55815905293563325856043959559, 7.83873337026315190808385447530, 8.317257046262649002426373369172, 8.897255014354849941293596911901, 9.453642667954677657666806461203, 9.651121492418243418918015037962, 10.23495964714052628902466770313, 10.37202673827620200804606497013

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