Properties

Label 2-152e2-1.1-c1-0-21
Degree $2$
Conductor $23104$
Sign $1$
Analytic cond. $184.486$
Root an. cond. $13.5825$
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  + 3·3-s − 2·5-s + 3·7-s + 6·9-s − 2·11-s + 3·13-s − 6·15-s − 17-s + 9·21-s − 5·23-s − 25-s + 9·27-s + 3·29-s + 6·31-s − 6·33-s − 6·35-s − 6·37-s + 9·39-s + 12·41-s − 10·43-s − 12·45-s + 8·47-s + 2·49-s − 3·51-s + 3·53-s + 4·55-s + 3·59-s + ⋯
L(s)  = 1  + 1.73·3-s − 0.894·5-s + 1.13·7-s + 2·9-s − 0.603·11-s + 0.832·13-s − 1.54·15-s − 0.242·17-s + 1.96·21-s − 1.04·23-s − 1/5·25-s + 1.73·27-s + 0.557·29-s + 1.07·31-s − 1.04·33-s − 1.01·35-s − 0.986·37-s + 1.44·39-s + 1.87·41-s − 1.52·43-s − 1.78·45-s + 1.16·47-s + 2/7·49-s − 0.420·51-s + 0.412·53-s + 0.539·55-s + 0.390·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23104\)    =    \(2^{6} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(184.486\)
Root analytic conductor: \(13.5825\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{23104} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 23104,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.621376773\)
\(L(\frac12)\) \(\approx\) \(4.621376773\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 - p T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 15 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−15.50050186188218, −14.83587053302646, −14.46248176414508, −13.82501346516334, −13.59058996799080, −12.94612978053702, −12.16676907471970, −11.77751476027279, −11.04517745675089, −10.51280718151187, −9.912230191082893, −9.214621596518936, −8.516905139809106, −8.219978523401588, −7.924841193688238, −7.349098253796952, −6.643904370339539, −5.727157060330037, −4.848525724286431, −4.229380649520713, −3.810893729181092, −3.109043574484374, −2.330594802281177, −1.787000267540175, −0.7947628208437609, 0.7947628208437609, 1.787000267540175, 2.330594802281177, 3.109043574484374, 3.810893729181092, 4.229380649520713, 4.848525724286431, 5.727157060330037, 6.643904370339539, 7.349098253796952, 7.924841193688238, 8.219978523401588, 8.516905139809106, 9.214621596518936, 9.912230191082893, 10.51280718151187, 11.04517745675089, 11.77751476027279, 12.16676907471970, 12.94612978053702, 13.59058996799080, 13.82501346516334, 14.46248176414508, 14.83587053302646, 15.50050186188218

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