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Properties

Label 2-186-31.5-c1-0-4
Degree $2$
Conductor $186$
Sign $-0.0336 + 0.999i$
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  − 2-s + (0.5 − 0.866i)3-s + 4-s + (−1 − 1.73i)5-s + (−0.5 + 0.866i)6-s + (0.5 − 0.866i)7-s − 8-s + (−0.499 − 0.866i)9-s + (1 + 1.73i)10-s + (−1.5 − 2.59i)11-s + (0.5 − 0.866i)12-s + (−0.5 + 0.866i)14-s − 1.99·15-s + 16-s + (1 − 1.73i)17-s + (0.499 + 0.866i)18-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.288 − 0.499i)3-s + 0.5·4-s + (−0.447 − 0.774i)5-s + (−0.204 + 0.353i)6-s + (0.188 − 0.327i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.316 + 0.547i)10-s + (−0.452 − 0.783i)11-s + (0.144 − 0.249i)12-s + (−0.133 + 0.231i)14-s − 0.516·15-s + 0.250·16-s + (0.242 − 0.420i)17-s + (0.117 + 0.204i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 186 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0336 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0336 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(186\)    =    \(2 \cdot 3 \cdot 31\)
Sign: $-0.0336 + 0.999i$
Analytic conductor: \(1.48521\)
Root analytic conductor: \(1.21869\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{186} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 186,\ (\ :1/2),\ -0.0336 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.575023 - 0.594737i\)
\(L(\frac12)\) \(\approx\) \(0.575023 - 0.594737i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (-5.5 - 0.866i)T \)
good5 \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - T + 29T^{2} \)
37 \( 1 + (4 - 6.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6 - 10.3i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2T + 47T^{2} \)
53 \( 1 + (-5.5 - 9.52i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 + (6 + 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4 - 6.92i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8 + 13.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.5 + 9.52i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 13T + 97T^{2} \)
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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

−12.21763544383976286729633957311, −11.42377480104219494118947323786, −10.31240066236971583950187066428, −9.098281142134381652068806066735, −8.264257156864171869084487029135, −7.55071717310242492544662252147, −6.24961365787797431802097801420, −4.73654110686147613734853210494, −2.96324101856691697297794933199, −0.955702912291211768812930275747, 2.38137687018075358794186271103, 3.80467402775441427998292768060, 5.43947792065295783103648066226, 6.94894955269803767596344740780, 7.83839829602055826644225828930, 8.848844777725781948786964654230, 9.990397400618930634315670827914, 10.64145535038080118737115280129, 11.64235420535508971106393877351, 12.61945966651026058468434914468

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