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Properties

Label 2-186-31.2-c1-0-3
Degree $2$
Conductor $186$
Sign $-0.525 + 0.850i$
Analytic cond. $1.48521$
Root an. cond. $1.21869$
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  + (−0.809 − 0.587i)2-s + (0.809 − 0.587i)3-s + (0.309 + 0.951i)4-s − 2.51·5-s − 6-s + (−1.13 − 3.50i)7-s + (0.309 − 0.951i)8-s + (0.309 − 0.951i)9-s + (2.03 + 1.47i)10-s + (−1.39 − 4.29i)11-s + (0.809 + 0.587i)12-s + (3.15 − 2.28i)13-s + (−1.13 + 3.50i)14-s + (−2.03 + 1.47i)15-s + (−0.809 + 0.587i)16-s + (−1.17 + 3.60i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (0.467 − 0.339i)3-s + (0.154 + 0.475i)4-s − 1.12·5-s − 0.408·6-s + (−0.430 − 1.32i)7-s + (0.109 − 0.336i)8-s + (0.103 − 0.317i)9-s + (0.643 + 0.467i)10-s + (−0.420 − 1.29i)11-s + (0.233 + 0.169i)12-s + (0.874 − 0.635i)13-s + (−0.304 + 0.936i)14-s + (−0.525 + 0.381i)15-s + (−0.202 + 0.146i)16-s + (−0.284 + 0.874i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 186 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(186\)    =    \(2 \cdot 3 \cdot 31\)
Sign: $-0.525 + 0.850i$
Analytic conductor: \(1.48521\)
Root analytic conductor: \(1.21869\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{186} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 186,\ (\ :1/2),\ -0.525 + 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.361242 - 0.647799i\)
\(L(\frac12)\) \(\approx\) \(0.361242 - 0.647799i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 + 0.587i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
31 \( 1 + (5.11 - 2.18i)T \)
good5 \( 1 + 2.51T + 5T^{2} \)
7 \( 1 + (1.13 + 3.50i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (1.39 + 4.29i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-3.15 + 2.28i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.17 - 3.60i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-2.53 - 1.84i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-0.618 + 1.90i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-0.256 - 0.186i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 - 4.49T + 37T^{2} \)
41 \( 1 + (-4.96 - 3.60i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-6.18 - 4.49i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-10.0 + 7.28i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.926 - 2.85i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (2.36 - 1.71i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + 12.3T + 61T^{2} \)
67 \( 1 - 11.2T + 67T^{2} \)
71 \( 1 + (-4.98 + 15.3i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (4.73 + 14.5i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-2.17 + 6.67i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-11.7 - 8.50i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-1.74 - 5.38i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-1.69 - 5.20i)T + (-78.4 + 57.0i)T^{2} \)
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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

−12.25557406048481986480289435649, −10.91236475746571209256347240893, −10.66536589427406601932800567753, −9.118695075612254780247718654055, −8.031531366678857097525010406532, −7.60768986725259125905008816570, −6.21197352538934304851815620182, −3.95357947876705760337689410673, −3.26773306418212533483680743775, −0.78648432637004769790946664245, 2.51417052047615038558447497056, 4.17475754453154156491257961553, 5.53612804351297327345536544870, 7.05384973844365427184415161578, 7.87567310643189095445304293895, 9.087010810051021256385136777905, 9.461664989038125939181680075171, 10.97963371401815780885231675290, 11.83791221570302928136979782045, 12.79883992734302973007905765228

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