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Properties

Label 2-2151-2151.238-c0-0-13
Degree $2$
Conductor $2151$
Sign $-0.882 + 0.469i$
Analytic cond. $1.07348$
Root an. cond. $1.03609$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.559 − 0.968i)2-s + (0.913 − 0.406i)3-s + (−0.125 + 0.217i)4-s + (0.719 − 1.24i)5-s + (−0.904 − 0.657i)6-s − 0.837·8-s + (0.669 − 0.743i)9-s − 1.60·10-s + (0.882 + 1.52i)11-s + (−0.0262 + 0.249i)12-s + (0.150 − 1.43i)15-s + (0.593 + 1.02i)16-s − 1.87·17-s + (−1.09 − 0.232i)18-s + (0.180 + 0.312i)20-s + ⋯
L(s)  = 1  + (−0.559 − 0.968i)2-s + (0.913 − 0.406i)3-s + (−0.125 + 0.217i)4-s + (0.719 − 1.24i)5-s + (−0.904 − 0.657i)6-s − 0.837·8-s + (0.669 − 0.743i)9-s − 1.60·10-s + (0.882 + 1.52i)11-s + (−0.0262 + 0.249i)12-s + (0.150 − 1.43i)15-s + (0.593 + 1.02i)16-s − 1.87·17-s + (−1.09 − 0.232i)18-s + (0.180 + 0.312i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.882 + 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.882 + 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2151\)    =    \(3^{2} \cdot 239\)
Sign: $-0.882 + 0.469i$
Analytic conductor: \(1.07348\)
Root analytic conductor: \(1.03609\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2151} (238, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2151,\ (\ :0),\ -0.882 + 0.469i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.351017016\)
\(L(\frac12)\) \(\approx\) \(1.351017016\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.913 + 0.406i)T \)
239 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (0.559 + 0.968i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.719 + 1.24i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.882 - 1.52i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + 1.87T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.990 + 1.71i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.997 + 1.72i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - 0.618T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.615 - 1.06i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−9.074414344730194400395686495481, −8.614938489689597298905845780869, −7.63088433137655421991834393250, −6.61984516582613379738674943950, −5.93245664291958416237991128514, −4.50874685368903783486580084607, −4.05106093178787892070921336508, −2.30945806659477409260704465565, −2.12143919687652681176087957271, −1.05986225903764420764233468530, 1.96572288657831685437620622318, 3.11316743310051897318152575735, 3.47944618499770605831596990321, 4.94347831846104095830507813923, 6.08463188820226165367749785126, 6.70242280877291376818905449965, 7.08455430012716520501591862539, 8.223106376251016510978222542127, 8.823129678087060018343394305075, 9.178312158704276795696348795450

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