Properties

Label 2-236992-1.1-c1-0-14
Degree $2$
Conductor $236992$
Sign $1$
Analytic cond. $1892.39$
Root an. cond. $43.5016$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 4·5-s + 7-s − 2·9-s − 2·13-s − 4·15-s + 5·17-s − 7·19-s − 21-s + 11·25-s + 5·27-s − 8·29-s + 2·31-s + 4·35-s − 8·37-s + 2·39-s − 2·41-s − 3·43-s − 8·45-s + 49-s − 5·51-s + 6·53-s + 7·57-s − 9·59-s − 6·61-s − 2·63-s − 8·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.78·5-s + 0.377·7-s − 2/3·9-s − 0.554·13-s − 1.03·15-s + 1.21·17-s − 1.60·19-s − 0.218·21-s + 11/5·25-s + 0.962·27-s − 1.48·29-s + 0.359·31-s + 0.676·35-s − 1.31·37-s + 0.320·39-s − 0.312·41-s − 0.457·43-s − 1.19·45-s + 1/7·49-s − 0.700·51-s + 0.824·53-s + 0.927·57-s − 1.17·59-s − 0.768·61-s − 0.251·63-s − 0.992·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(236992\)    =    \(2^{6} \cdot 7 \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(1892.39\)
Root analytic conductor: \(43.5016\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{236992} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 236992,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.005710386\)
\(L(\frac12)\) \(\approx\) \(2.005710386\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - T \)
23 \( 1 \)
good3 \( 1 + T + p T^{2} \)
5 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 3 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 17 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 13 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.81123117510549, −12.43283581162857, −12.14060400131971, −11.40483316083505, −10.98698259960325, −10.51193768300344, −10.21910645000864, −9.680259930899124, −9.257665960496069, −8.741094765771376, −8.298593063961163, −7.753330360720732, −6.961286478499953, −6.674049623820811, −6.027426800300022, −5.739881219184479, −5.254570393167260, −4.972007933945969, −4.258763544656289, −3.466625277917033, −2.899762792420467, −2.218265489175408, −1.876846310958820, −1.266172731594957, −0.3908536549467475, 0.3908536549467475, 1.266172731594957, 1.876846310958820, 2.218265489175408, 2.899762792420467, 3.466625277917033, 4.258763544656289, 4.972007933945969, 5.254570393167260, 5.739881219184479, 6.027426800300022, 6.674049623820811, 6.961286478499953, 7.753330360720732, 8.298593063961163, 8.741094765771376, 9.257665960496069, 9.680259930899124, 10.21910645000864, 10.51193768300344, 10.98698259960325, 11.40483316083505, 12.14060400131971, 12.43283581162857, 12.81123117510549

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