Properties

Label 4-546e2-1.1-c1e2-0-33
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3·3-s + 3·4-s − 6·6-s + 5·7-s + 4·8-s + 6·9-s − 9·12-s − 2·13-s + 10·14-s + 5·16-s − 6·17-s + 12·18-s − 4·19-s − 15·21-s − 12·24-s + 7·25-s − 4·26-s − 9·27-s + 15·28-s + 16·31-s + 6·32-s − 12·34-s + 18·36-s − 8·38-s + 6·39-s − 30·42-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.73·3-s + 3/2·4-s − 2.44·6-s + 1.88·7-s + 1.41·8-s + 2·9-s − 2.59·12-s − 0.554·13-s + 2.67·14-s + 5/4·16-s − 1.45·17-s + 2.82·18-s − 0.917·19-s − 3.27·21-s − 2.44·24-s + 7/5·25-s − 0.784·26-s − 1.73·27-s + 2.83·28-s + 2.87·31-s + 1.06·32-s − 2.05·34-s + 3·36-s − 1.29·38-s + 0.960·39-s − 4.62·42-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: \(0\)
Selberg data: \((4,\ 298116,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.064127008\)
\(L(\frac12)\) \(\approx\) \(3.064127008\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T + p T^{2} \)
7$C_2$ \( 1 - 5 T + p T^{2} \)
13$C_2$ \( 1 + 2 T + p T^{2} \)
good5$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 71 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 53 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 154 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 166 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
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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

−11.12527416843747569924736155900, −10.81819987192700384126653738571, −10.51081536822856523012362329335, −10.08642791535168582302754099414, −9.446308119477983334854180644566, −8.488621579908768185943012188625, −8.300422109758970489867340675834, −7.85036162334922702114329437952, −6.89889442190176206024077455638, −6.77748434292119410393062732674, −6.50379392837217768899571940823, −5.83743061463752645412166003067, −5.19261235971330291783398588582, −4.85767348395588746651493645720, −4.62146132573888650910578520334, −4.37952033965708351220728726435, −3.42847535983241504989019650171, −2.39770062202245478712646422186, −1.91088873287514032072613703014, −0.956148636789177514059356906960, 0.956148636789177514059356906960, 1.91088873287514032072613703014, 2.39770062202245478712646422186, 3.42847535983241504989019650171, 4.37952033965708351220728726435, 4.62146132573888650910578520334, 4.85767348395588746651493645720, 5.19261235971330291783398588582, 5.83743061463752645412166003067, 6.50379392837217768899571940823, 6.77748434292119410393062732674, 6.89889442190176206024077455638, 7.85036162334922702114329437952, 8.300422109758970489867340675834, 8.488621579908768185943012188625, 9.446308119477983334854180644566, 10.08642791535168582302754099414, 10.51081536822856523012362329335, 10.81819987192700384126653738571, 11.12527416843747569924736155900

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