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Properties

Label 2-2151-2151.238-c0-0-11
Degree $2$
Conductor $2151$
Sign $0.438 - 0.898i$
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  + (0.615 + 1.06i)2-s + (0.669 + 0.743i)3-s + (−0.258 + 0.447i)4-s + (0.882 − 1.52i)5-s + (−0.380 + 1.17i)6-s + 0.595·8-s + (−0.104 + 0.994i)9-s + 2.17·10-s + (−0.438 − 0.759i)11-s + (−0.504 + 0.107i)12-s + (1.72 − 0.367i)15-s + (0.624 + 1.08i)16-s − 1.87·17-s + (−1.12 + 0.500i)18-s + (0.455 + 0.789i)20-s + ⋯
L(s)  = 1  + (0.615 + 1.06i)2-s + (0.669 + 0.743i)3-s + (−0.258 + 0.447i)4-s + (0.882 − 1.52i)5-s + (−0.380 + 1.17i)6-s + 0.595·8-s + (−0.104 + 0.994i)9-s + 2.17·10-s + (−0.438 − 0.759i)11-s + (−0.504 + 0.107i)12-s + (1.72 − 0.367i)15-s + (0.624 + 1.08i)16-s − 1.87·17-s + (−1.12 + 0.500i)18-s + (0.455 + 0.789i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.438 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.438 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2151\)    =    \(3^{2} \cdot 239\)
Sign: $0.438 - 0.898i$
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: \((2,\ 2151,\ (\ :0),\ 0.438 - 0.898i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.348537333\)
\(L(\frac12)\) \(\approx\) \(2.348537333\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.669 - 0.743i)T \)
239 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.615 - 1.06i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.882 + 1.52i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.438 + 0.759i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (-0.719 - 1.24i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.374 + 0.648i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + 1.61T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.961 + 1.66i)T + (-0.5 + 0.866i)T^{2} \)
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

−8.951221113099828380058817863249, −8.747484627881923774419016206702, −8.021243174066405914221048813284, −6.97368187142172889708061534355, −5.95788632012803540745673108641, −5.44464369071671900942615112725, −4.58207564936077763430385392647, −4.29516351890170091740020258394, −2.75040765097142924306720339971, −1.58506419278609456547249467214, 1.82403661199129300891115171628, 2.36855743433133820284109334828, 2.91590647046062582673683102804, 3.88712924140766114705937483677, 4.89219376475542144459084135068, 6.22374478681585136111586697701, 6.76603293962416093889558857966, 7.40684336655687804531984093870, 8.310544724810099415954396907950, 9.425306115610237941140024290235

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