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Properties

Label 2-2151-2151.238-c0-0-4
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  + (0.997 + 1.72i)2-s + (0.913 − 0.406i)3-s + (−1.49 + 2.58i)4-s + (−0.961 + 1.66i)5-s + (1.61 + 1.17i)6-s − 3.95·8-s + (0.669 − 0.743i)9-s − 3.83·10-s + (−0.0348 − 0.0604i)11-s + (−0.311 + 2.96i)12-s + (−0.200 + 1.91i)15-s + (−2.45 − 4.24i)16-s + 1.53·17-s + (1.95 + 0.414i)18-s + (−2.86 − 4.96i)20-s + ⋯
L(s)  = 1  + (0.997 + 1.72i)2-s + (0.913 − 0.406i)3-s + (−1.49 + 2.58i)4-s + (−0.961 + 1.66i)5-s + (1.61 + 1.17i)6-s − 3.95·8-s + (0.669 − 0.743i)9-s − 3.83·10-s + (−0.0348 − 0.0604i)11-s + (−0.311 + 2.96i)12-s + (−0.200 + 1.91i)15-s + (−2.45 − 4.24i)16-s + 1.53·17-s + (1.95 + 0.414i)18-s + (−2.86 − 4.96i)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

\(L(\frac{1}{2})\) \(\approx\) \(1.903145230\)
\(L(\frac12)\) \(\approx\) \(1.903145230\)
\(L(1)\) 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 \( 1 + (-0.997 - 1.72i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.961 - 1.66i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.0348 + 0.0604i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 - 1.53T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.615 - 1.06i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.438 - 0.759i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - 0.618T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.374 - 0.648i)T + (-0.5 + 0.866i)T^{2} \)
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

−9.303329080244537979034085977628, −8.363452032286710789382115014682, −7.79468160727189589265718466685, −7.36417807000800852169474896306, −6.71582124411464513653764038329, −6.14974029740198166470881500809, −4.98649458929666560406169179121, −3.88615819688766846719620617227, −3.33766854410926899591712544174, −2.82205324768679541768093978682, 0.930489967859464203128246875586, 1.93922319642535975971852846479, 3.15891511283702407713870450950, 3.83046993276193204253550744921, 4.46402423195189368566487599290, 5.07159656160562288876180001232, 5.84064625920859818589488247120, 7.64839228921540445146990920607, 8.422666608446199853248736454639, 9.054968805107860182458687265785

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