Properties

Label 4-546e2-1.1-c1e2-0-22
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-s − 3-s + 5-s + 6-s − 7-s + 8-s − 10-s + 11-s − 2·13-s + 14-s − 15-s − 16-s + 6·17-s + 4·19-s + 21-s − 22-s + 6·23-s − 24-s + 5·25-s + 2·26-s + 27-s + 6·29-s + 30-s + 11·31-s − 33-s − 6·34-s − 35-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.447·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s − 0.316·10-s + 0.301·11-s − 0.554·13-s + 0.267·14-s − 0.258·15-s − 1/4·16-s + 1.45·17-s + 0.917·19-s + 0.218·21-s − 0.213·22-s + 1.25·23-s − 0.204·24-s + 25-s + 0.392·26-s + 0.192·27-s + 1.11·29-s + 0.182·30-s + 1.97·31-s − 0.174·33-s − 1.02·34-s − 0.169·35-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: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 298116,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.154056637\)
\(L(\frac12)\) \(\approx\) \(1.154056637\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + p T^{2} \)
13$C_1$ \( ( 1 + T )^{2} \)
good5$C_2^2$ \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
37$C_2^2$ \( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 5 T - 28 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 5 T - 34 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 12 T + 83 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
79$C_2^2$ \( 1 + 7 T - 30 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 17 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 12 T + 55 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \)
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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

−10.79292762452724638664465111222, −10.59968536796956491800518294361, −9.941274010020743059097259597581, −9.820929896446834318321933931817, −9.399962436611779990352195139564, −8.867364660409077181261747633233, −8.440177060108751622383883953505, −7.969137781154364086106483468282, −7.22440246136445821184965402644, −7.22346425429358204649414479361, −6.42425224506268994397092301196, −6.05502022869840190829460552562, −5.53074109270997193839021520966, −4.84520803212155032273063532775, −4.71654687659452427119489782948, −3.76773780067947046869286477055, −2.83931838297286137220896364174, −2.78934340961851449924498546850, −1.29709934570008451130050918403, −0.871446442840109719682469124242, 0.871446442840109719682469124242, 1.29709934570008451130050918403, 2.78934340961851449924498546850, 2.83931838297286137220896364174, 3.76773780067947046869286477055, 4.71654687659452427119489782948, 4.84520803212155032273063532775, 5.53074109270997193839021520966, 6.05502022869840190829460552562, 6.42425224506268994397092301196, 7.22346425429358204649414479361, 7.22440246136445821184965402644, 7.969137781154364086106483468282, 8.440177060108751622383883953505, 8.867364660409077181261747633233, 9.399962436611779990352195139564, 9.820929896446834318321933931817, 9.941274010020743059097259597581, 10.59968536796956491800518294361, 10.79292762452724638664465111222

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