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Properties

Label 2-187-17.4-c1-0-2
Degree $2$
Conductor $187$
Sign $0.930 - 0.366i$
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  − 1.90i·2-s + (−1.14 + 1.14i)3-s − 1.62·4-s + (−2.29 + 2.29i)5-s + (2.17 + 2.17i)6-s + (3.32 + 3.32i)7-s − 0.709i·8-s + 0.399i·9-s + (4.36 + 4.36i)10-s + (−0.707 − 0.707i)11-s + (1.85 − 1.85i)12-s + 2.85·13-s + (6.34 − 6.34i)14-s − 5.22i·15-s − 4.60·16-s + (−3.62 + 1.96i)17-s + ⋯
L(s)  = 1  − 1.34i·2-s + (−0.658 + 0.658i)3-s − 0.813·4-s + (−1.02 + 1.02i)5-s + (0.886 + 0.886i)6-s + (1.25 + 1.25i)7-s − 0.250i·8-s + 0.133i·9-s + (1.38 + 1.38i)10-s + (−0.213 − 0.213i)11-s + (0.535 − 0.535i)12-s + 0.791·13-s + (1.69 − 1.69i)14-s − 1.34i·15-s − 1.15·16-s + (−0.878 + 0.476i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 - 0.366i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.930 - 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $0.930 - 0.366i$
Analytic conductor: \(1.49320\)
Root analytic conductor: \(1.22196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 187,\ (\ :1/2),\ 0.930 - 0.366i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.841177 + 0.159471i\)
\(L(\frac12)\) \(\approx\) \(0.841177 + 0.159471i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.62 - 1.96i)T \)
good2 \( 1 + 1.90iT - 2T^{2} \)
3 \( 1 + (1.14 - 1.14i)T - 3iT^{2} \)
5 \( 1 + (2.29 - 2.29i)T - 5iT^{2} \)
7 \( 1 + (-3.32 - 3.32i)T + 7iT^{2} \)
13 \( 1 - 2.85T + 13T^{2} \)
19 \( 1 - 3.83iT - 19T^{2} \)
23 \( 1 + (-3.28 - 3.28i)T + 23iT^{2} \)
29 \( 1 + (-3.14 + 3.14i)T - 29iT^{2} \)
31 \( 1 + (-1.43 + 1.43i)T - 31iT^{2} \)
37 \( 1 + (-1.52 + 1.52i)T - 37iT^{2} \)
41 \( 1 + (-5.52 - 5.52i)T + 41iT^{2} \)
43 \( 1 + 12.7iT - 43T^{2} \)
47 \( 1 + 5.54T + 47T^{2} \)
53 \( 1 + 4.51iT - 53T^{2} \)
59 \( 1 + 0.104iT - 59T^{2} \)
61 \( 1 + (-3.14 - 3.14i)T + 61iT^{2} \)
67 \( 1 + 2.20T + 67T^{2} \)
71 \( 1 + (-3.31 + 3.31i)T - 71iT^{2} \)
73 \( 1 + (-2.33 + 2.33i)T - 73iT^{2} \)
79 \( 1 + (4.15 + 4.15i)T + 79iT^{2} \)
83 \( 1 + 15.1iT - 83T^{2} \)
89 \( 1 - 18.3T + 89T^{2} \)
97 \( 1 + (3.71 - 3.71i)T - 97iT^{2} \)
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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

−11.96831149342838338624745276142, −11.45586959587126702186438331327, −11.00215557333200901811107516690, −10.28667661717074202989463151018, −8.819729750496429056804650574136, −7.77979194365888000395148415578, −6.09390524996683481586456543529, −4.71926650331543926002509011708, −3.60275575964924863083575997768, −2.18222217030183084337722211746, 0.896547421737418930582258290921, 4.40583461612377421874355772396, 4.99925610626819659635969018992, 6.55550078996858972059114017292, 7.30966873371866068017172083006, 8.119847339404515475871704310209, 8.903868232254994056006343326709, 11.07047560233223341080267614294, 11.40316184235823303519994518345, 12.67962667364860509911894778076

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