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Properties

Label 2-182-13.3-c1-0-4
Degree $2$
Conductor $182$
Sign $-0.189 + 0.981i$
Analytic cond. $1.45327$
Root an. cond. $1.20551$
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  + (−0.5 − 0.866i)2-s + (−0.866 − 1.5i)3-s + (−0.499 + 0.866i)4-s + 3.73·5-s + (−0.866 + 1.5i)6-s + (0.5 − 0.866i)7-s + 0.999·8-s + (−1.86 − 3.23i)10-s + (−0.732 − 1.26i)11-s + 1.73·12-s + (−3.59 + 0.232i)13-s − 0.999·14-s + (−3.23 − 5.59i)15-s + (−0.5 − 0.866i)16-s + (−1.73 + 3i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.499 − 0.866i)3-s + (−0.249 + 0.433i)4-s + 1.66·5-s + (−0.353 + 0.612i)6-s + (0.188 − 0.327i)7-s + 0.353·8-s + (−0.590 − 1.02i)10-s + (−0.220 − 0.382i)11-s + 0.499·12-s + (−0.997 + 0.0643i)13-s − 0.267·14-s + (−0.834 − 1.44i)15-s + (−0.125 − 0.216i)16-s + (−0.420 + 0.727i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.189 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(182\)    =    \(2 \cdot 7 \cdot 13\)
Sign: $-0.189 + 0.981i$
Analytic conductor: \(1.45327\)
Root analytic conductor: \(1.20551\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{182} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 182,\ (\ :1/2),\ -0.189 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.658964 - 0.798572i\)
\(L(\frac12)\) \(\approx\) \(0.658964 - 0.798572i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (3.59 - 0.232i)T \)
good3 \( 1 + (0.866 + 1.5i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 3.73T + 5T^{2} \)
11 \( 1 + (0.732 + 1.26i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.73 - 3i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.73 + 3i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.767 + 1.33i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.46 - 4.26i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.46T + 31T^{2} \)
37 \( 1 + (-4.73 - 8.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.26 + 2.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2 - 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 8.92T + 47T^{2} \)
53 \( 1 - 6.92T + 53T^{2} \)
59 \( 1 + (6.59 - 11.4i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.59 + 6.23i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.46 - 7.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.23 - 3.86i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 7.46T + 73T^{2} \)
79 \( 1 - 14.9T + 79T^{2} \)
83 \( 1 + 10.3T + 83T^{2} \)
89 \( 1 + (-7.73 - 13.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.46 + 14.6i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.41865404452316843677242589980, −11.41369142524256575891728682887, −10.27728079260429162640046546258, −9.652387534121871558098379380131, −8.425272307194932629562653253830, −7.01003093476848975765054804417, −6.16072386860696360376475586974, −4.86387371063392812107940838653, −2.62651341964235625470450629982, −1.30882997859025785313984432881, 2.22761269501923328522118226249, 4.74468526134738212759754742199, 5.41748509604406618311149688849, 6.41996483096302888910295124953, 7.79011939500814781857554289919, 9.314322439047034354674940831990, 9.837847123192750224180854821099, 10.46405615099176482202146886457, 11.79536417653894881923766541087, 13.15106282439260736798845508710

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