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Properties

Label 2-187-17.13-c1-0-9
Degree $2$
Conductor $187$
Sign $0.795 - 0.606i$
Analytic cond. $1.49320$
Root an. cond. $1.22196$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.385i·2-s + (1.16 + 1.16i)3-s + 1.85·4-s + (−0.671 − 0.671i)5-s + (−0.448 + 0.448i)6-s + (1.38 − 1.38i)7-s + 1.48i·8-s − 0.283i·9-s + (0.258 − 0.258i)10-s + (−0.707 + 0.707i)11-s + (2.15 + 2.15i)12-s − 5.16·13-s + (0.533 + 0.533i)14-s − 1.56i·15-s + 3.13·16-s + (−3.56 + 2.07i)17-s + ⋯
L(s)  = 1  + 0.272i·2-s + (0.672 + 0.672i)3-s + 0.925·4-s + (−0.300 − 0.300i)5-s + (−0.183 + 0.183i)6-s + (0.524 − 0.524i)7-s + 0.524i·8-s − 0.0946i·9-s + (0.0817 − 0.0817i)10-s + (−0.213 + 0.213i)11-s + (0.622 + 0.622i)12-s − 1.43·13-s + (0.142 + 0.142i)14-s − 0.404i·15-s + 0.783·16-s + (−0.863 + 0.504i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.795 - 0.606i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.795 - 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $0.795 - 0.606i$
Analytic conductor: \(1.49320\)
Root analytic conductor: \(1.22196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (166, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 187,\ (\ :1/2),\ 0.795 - 0.606i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52176 + 0.513691i\)
\(L(\frac12)\) \(\approx\) \(1.52176 + 0.513691i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 + (3.56 - 2.07i)T \)
good2 \( 1 - 0.385iT - 2T^{2} \)
3 \( 1 + (-1.16 - 1.16i)T + 3iT^{2} \)
5 \( 1 + (0.671 + 0.671i)T + 5iT^{2} \)
7 \( 1 + (-1.38 + 1.38i)T - 7iT^{2} \)
13 \( 1 + 5.16T + 13T^{2} \)
19 \( 1 - 1.52iT - 19T^{2} \)
23 \( 1 + (3.45 - 3.45i)T - 23iT^{2} \)
29 \( 1 + (-0.706 - 0.706i)T + 29iT^{2} \)
31 \( 1 + (-2.25 - 2.25i)T + 31iT^{2} \)
37 \( 1 + (0.724 + 0.724i)T + 37iT^{2} \)
41 \( 1 + (-0.427 + 0.427i)T - 41iT^{2} \)
43 \( 1 + 12.0iT - 43T^{2} \)
47 \( 1 + 5.76T + 47T^{2} \)
53 \( 1 + 7.56iT - 53T^{2} \)
59 \( 1 + 3.89iT - 59T^{2} \)
61 \( 1 + (5.57 - 5.57i)T - 61iT^{2} \)
67 \( 1 - 10.8T + 67T^{2} \)
71 \( 1 + (-5.20 - 5.20i)T + 71iT^{2} \)
73 \( 1 + (-6.00 - 6.00i)T + 73iT^{2} \)
79 \( 1 + (2.85 - 2.85i)T - 79iT^{2} \)
83 \( 1 + 2.51iT - 83T^{2} \)
89 \( 1 - 17.8T + 89T^{2} \)
97 \( 1 + (-11.0 - 11.0i)T + 97iT^{2} \)
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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

−12.48913632857290615620626829235, −11.75652895771972337078940000687, −10.56444926633682316570137720151, −9.818866904692511248595843962233, −8.492713720169484237537880158031, −7.67705740279481446732302142618, −6.61324662260584263030439713921, −5.02363485291468543801034270832, −3.84157216001326644287035609661, −2.27928946772903070654914668509, 2.09705185437740968963978660392, 2.87516567401443428184774187117, 4.87745500940818021965490085249, 6.49974436076311816726953199375, 7.47474709909442357308917367068, 8.137799762366915084934048726422, 9.500480787010195101232256033086, 10.75417977006704046648571069502, 11.57229783677571689755698090171, 12.38454778669238103439962759682

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