Properties

Label 2-187-17.13-c1-0-5
Degree $2$
Conductor $187$
Sign $0.933 - 0.358i$
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  − 0.0421i·2-s + (−0.440 − 0.440i)3-s + 1.99·4-s + (2.27 + 2.27i)5-s + (−0.0185 + 0.0185i)6-s + (−3.03 + 3.03i)7-s − 0.168i·8-s − 2.61i·9-s + (0.0957 − 0.0957i)10-s + (−0.707 + 0.707i)11-s + (−0.880 − 0.880i)12-s + 3.32·13-s + (0.127 + 0.127i)14-s − 2.00i·15-s + 3.98·16-s + (−4.00 + 0.994i)17-s + ⋯
L(s)  = 1  − 0.0297i·2-s + (−0.254 − 0.254i)3-s + 0.999·4-s + (1.01 + 1.01i)5-s + (−0.00757 + 0.00757i)6-s + (−1.14 + 1.14i)7-s − 0.0595i·8-s − 0.870i·9-s + (0.0302 − 0.0302i)10-s + (−0.213 + 0.213i)11-s + (−0.254 − 0.254i)12-s + 0.921·13-s + (0.0341 + 0.0341i)14-s − 0.517i·15-s + 0.997·16-s + (−0.970 + 0.241i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 - 0.358i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 - 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $0.933 - 0.358i$
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: \((2,\ 187,\ (\ :1/2),\ 0.933 - 0.358i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36274 + 0.252294i\)
\(L(\frac12)\) \(\approx\) \(1.36274 + 0.252294i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.00 - 0.994i)T \)
good2 \( 1 + 0.0421iT - 2T^{2} \)
3 \( 1 + (0.440 + 0.440i)T + 3iT^{2} \)
5 \( 1 + (-2.27 - 2.27i)T + 5iT^{2} \)
7 \( 1 + (3.03 - 3.03i)T - 7iT^{2} \)
13 \( 1 - 3.32T + 13T^{2} \)
19 \( 1 + 5.85iT - 19T^{2} \)
23 \( 1 + (-3.81 + 3.81i)T - 23iT^{2} \)
29 \( 1 + (-0.616 - 0.616i)T + 29iT^{2} \)
31 \( 1 + (6.28 + 6.28i)T + 31iT^{2} \)
37 \( 1 + (0.532 + 0.532i)T + 37iT^{2} \)
41 \( 1 + (4.20 - 4.20i)T - 41iT^{2} \)
43 \( 1 + 0.849iT - 43T^{2} \)
47 \( 1 + 8.83T + 47T^{2} \)
53 \( 1 - 3.41iT - 53T^{2} \)
59 \( 1 - 12.4iT - 59T^{2} \)
61 \( 1 + (-8.41 + 8.41i)T - 61iT^{2} \)
67 \( 1 - 4.87T + 67T^{2} \)
71 \( 1 + (2.47 + 2.47i)T + 71iT^{2} \)
73 \( 1 + (3.10 + 3.10i)T + 73iT^{2} \)
79 \( 1 + (4.40 - 4.40i)T - 79iT^{2} \)
83 \( 1 - 6.33iT - 83T^{2} \)
89 \( 1 - 0.373T + 89T^{2} \)
97 \( 1 + (-8.46 - 8.46i)T + 97iT^{2} \)
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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

−12.73971510397303935887994681243, −11.52960961905686885260200204325, −10.83055923890621133474578279953, −9.715479579770720264505729346777, −8.866197641133229630519023230616, −6.85568511051048808713096552625, −6.50430277352873804107862802148, −5.74970345413013094606372624402, −3.17900539535288025155111817514, −2.28448573214436620209050730179, 1.64552615609133119991374495530, 3.53472786914358508296033243486, 5.20729796813615427591745583250, 6.17687421600022848036668822864, 7.19803368530661204521116495757, 8.544689329470373912644933077953, 9.830376978209236452370715204375, 10.48649635326402913736982361851, 11.34567717833949190456032827953, 12.86308525817297096671119640674

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