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Properties

Label 2-2151-2151.1672-c0-0-6
Degree $2$
Conductor $2151$
Sign $0.990 - 0.139i$
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  + (0.241 − 0.419i)2-s + (−0.104 + 0.994i)3-s + (0.382 + 0.663i)4-s + (−0.559 − 0.968i)5-s + (0.391 + 0.284i)6-s + 0.854·8-s + (−0.978 − 0.207i)9-s − 0.541·10-s + (0.615 − 1.06i)11-s + (−0.699 + 0.311i)12-s + (1.02 − 0.454i)15-s + (−0.176 + 0.305i)16-s + 1.53·17-s + (−0.323 + 0.359i)18-s + (0.428 − 0.741i)20-s + ⋯
L(s)  = 1  + (0.241 − 0.419i)2-s + (−0.104 + 0.994i)3-s + (0.382 + 0.663i)4-s + (−0.559 − 0.968i)5-s + (0.391 + 0.284i)6-s + 0.854·8-s + (−0.978 − 0.207i)9-s − 0.541·10-s + (0.615 − 1.06i)11-s + (−0.699 + 0.311i)12-s + (1.02 − 0.454i)15-s + (−0.176 + 0.305i)16-s + 1.53·17-s + (−0.323 + 0.359i)18-s + (0.428 − 0.741i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2151\)    =    \(3^{2} \cdot 239\)
Sign: $0.990 - 0.139i$
Analytic conductor: \(1.07348\)
Root analytic conductor: \(1.03609\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2151} (1672, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2151,\ (\ :0),\ 0.990 - 0.139i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.400412569\)
\(L(\frac12)\) \(\approx\) \(1.400412569\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.104 - 0.994i)T \)
239 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-0.241 + 0.419i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.559 + 0.968i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.615 + 1.06i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (0.0348 - 0.0604i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.719 - 1.24i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (0.848 - 1.46i)T + (-0.5 - 0.866i)T^{2} \)
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

−9.241542839480757969425734131204, −8.362789707473284261907003923665, −8.191592104299715091033920102140, −6.98309379246535699694066397994, −5.90408691343226257622475516301, −5.06853269361284621064766365606, −4.26695224321676501986254864040, −3.54250872388363261254357922434, −2.93845751922386689553334000120, −1.18858168631166570315654176987, 1.26861244754700612116583865488, 2.31146865425987117289953080825, 3.34729670815723695666026265703, 4.54607747705904831920166099458, 5.56121512828724268139361051344, 6.32367657608263968442128734134, 6.86079810019095776177969607163, 7.61354883585283147632832303879, 7.893396731594760217131470503134, 9.330988956419233261960271963169

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