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Properties

Label 2-2151-2151.1672-c0-0-1
Degree $2$
Conductor $2151$
Sign $-0.719 + 0.694i$
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  + (−0.848 + 1.46i)2-s + (−0.978 + 0.207i)3-s + (−0.938 − 1.62i)4-s + (0.615 + 1.06i)5-s + (0.524 − 1.61i)6-s + 1.48·8-s + (0.913 − 0.406i)9-s − 2.08·10-s + (−0.961 + 1.66i)11-s + (1.25 + 1.39i)12-s + (−0.823 − 0.915i)15-s + (−0.322 + 0.558i)16-s + 1.53·17-s + (−0.177 + 1.68i)18-s + (1.15 − 2.00i)20-s + ⋯
L(s)  = 1  + (−0.848 + 1.46i)2-s + (−0.978 + 0.207i)3-s + (−0.938 − 1.62i)4-s + (0.615 + 1.06i)5-s + (0.524 − 1.61i)6-s + 1.48·8-s + (0.913 − 0.406i)9-s − 2.08·10-s + (−0.961 + 1.66i)11-s + (1.25 + 1.39i)12-s + (−0.823 − 0.915i)15-s + (−0.322 + 0.558i)16-s + 1.53·17-s + (−0.177 + 1.68i)18-s + (1.15 − 2.00i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.719 + 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.719 + 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2151\)    =    \(3^{2} \cdot 239\)
Sign: $-0.719 + 0.694i$
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: \((2,\ 2151,\ (\ :0),\ -0.719 + 0.694i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4806894968\)
\(L(\frac12)\) \(\approx\) \(0.4806894968\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.978 - 0.207i)T \)
239 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.848 - 1.46i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (-0.615 - 1.06i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.961 - 1.66i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (0.559 - 0.968i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.882 - 1.52i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + 1.61T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.997 + 1.72i)T + (-0.5 - 0.866i)T^{2} \)
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

−9.876769095871119192170116367731, −9.014368935415700441430079078607, −7.82531956156001582804836719658, −7.24865600570339856968895787588, −6.80259880996327575468592637154, −5.99705101908468550591145048353, −5.30808487161393889945833113243, −4.69819917730419610263238844993, −3.09097197010890830241674388559, −1.55625351009818551393931590265, 0.57387094503591662576650116101, 1.35439261655419051424549219296, 2.53565269838349848531665520969, 3.59877128804782167935158648578, 4.71764842793354407409246845164, 5.67200563971677939819412299362, 6.04903428444449485108085916564, 7.82027368839699837772481697192, 8.055920971070819355543742606790, 9.082241864589511149186815733247

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