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Properties

Label 2-187-17.16-c1-0-8
Degree $2$
Conductor $187$
Sign $0.808 - 0.588i$
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  + 1.89·2-s + 1.04i·3-s + 1.58·4-s + 1.83i·5-s + 1.97i·6-s − 1.61i·7-s − 0.787·8-s + 1.91·9-s + 3.47i·10-s − i·11-s + 1.64i·12-s + 1.28·13-s − 3.06i·14-s − 1.91·15-s − 4.65·16-s + (−2.42 − 3.33i)17-s + ⋯
L(s)  = 1  + 1.33·2-s + 0.600i·3-s + 0.791·4-s + 0.821i·5-s + 0.804i·6-s − 0.611i·7-s − 0.278·8-s + 0.638·9-s + 1.09i·10-s − 0.301i·11-s + 0.475i·12-s + 0.355·13-s − 0.818i·14-s − 0.493·15-s − 1.16·16-s + (−0.588 − 0.808i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.808 - 0.588i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.808 - 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $0.808 - 0.588i$
Analytic conductor: \(1.49320\)
Root analytic conductor: \(1.22196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 187,\ (\ :1/2),\ 0.808 - 0.588i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.02232 + 0.658302i\)
\(L(\frac12)\) \(\approx\) \(2.02232 + 0.658302i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + iT \)
17 \( 1 + (2.42 + 3.33i)T \)
good2 \( 1 - 1.89T + 2T^{2} \)
3 \( 1 - 1.04iT - 3T^{2} \)
5 \( 1 - 1.83iT - 5T^{2} \)
7 \( 1 + 1.61iT - 7T^{2} \)
13 \( 1 - 1.28T + 13T^{2} \)
19 \( 1 + 3.44T + 19T^{2} \)
23 \( 1 + 4.87iT - 23T^{2} \)
29 \( 1 + 5.71iT - 29T^{2} \)
31 \( 1 - 3.11iT - 31T^{2} \)
37 \( 1 - 6.82iT - 37T^{2} \)
41 \( 1 - 4.07iT - 41T^{2} \)
43 \( 1 + 8.87T + 43T^{2} \)
47 \( 1 - 5.88T + 47T^{2} \)
53 \( 1 + 3.22T + 53T^{2} \)
59 \( 1 + 2.40T + 59T^{2} \)
61 \( 1 - 13.3iT - 61T^{2} \)
67 \( 1 - 5.72T + 67T^{2} \)
71 \( 1 - 6.48iT - 71T^{2} \)
73 \( 1 - 4.32iT - 73T^{2} \)
79 \( 1 + 15.1iT - 79T^{2} \)
83 \( 1 + 8.83T + 83T^{2} \)
89 \( 1 + 0.224T + 89T^{2} \)
97 \( 1 + 3.00iT - 97T^{2} \)
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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

−12.99002512522108694161544408017, −11.76928583882120794298005369303, −10.83862891398484790497570837217, −10.06284000806083344550767100303, −8.719512676726857133491545189195, −7.05568500644076739015254072440, −6.30528293552710812877190454614, −4.80009303611984447100443606261, −4.02957418763846889561359677161, −2.83776191786203429437057917443, 1.97423549472464444989826093792, 3.82079442698140655105277042188, 4.89036436388243281594176341394, 5.93793226652855795659352555188, 6.97304390193735269332308113399, 8.422404263822166197362138671547, 9.334309511637850707823186454491, 10.90527160803529920810488977910, 12.11396567712503709547977506130, 12.71269317967454583120364954475

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