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Properties

Label 2-182-91.81-c1-0-2
Degree $2$
Conductor $182$
Sign $0.976 - 0.213i$
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 − 2.39·3-s + (−0.499 − 0.866i)4-s + (−0.561 − 0.972i)5-s + (1.19 − 2.07i)6-s + (2.64 − 0.173i)7-s + 0.999·8-s + 2.75·9-s + 1.12·10-s + 5.57·11-s + (1.19 + 2.07i)12-s + (−0.197 + 3.60i)13-s + (−1.16 + 2.37i)14-s + (1.34 + 2.33i)15-s + (−0.5 + 0.866i)16-s + (−3.66 − 6.35i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s − 1.38·3-s + (−0.249 − 0.433i)4-s + (−0.251 − 0.434i)5-s + (0.489 − 0.848i)6-s + (0.997 − 0.0656i)7-s + 0.353·8-s + 0.919·9-s + 0.355·10-s + 1.68·11-s + (0.346 + 0.599i)12-s + (−0.0548 + 0.998i)13-s + (−0.312 + 0.634i)14-s + (0.347 + 0.602i)15-s + (−0.125 + 0.216i)16-s + (−0.889 − 1.54i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.213i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 - 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(182\)    =    \(2 \cdot 7 \cdot 13\)
Sign: $0.976 - 0.213i$
Analytic conductor: \(1.45327\)
Root analytic conductor: \(1.20551\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{182} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 182,\ (\ :1/2),\ 0.976 - 0.213i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.692387 + 0.0749269i\)
\(L(\frac12)\) \(\approx\) \(0.692387 + 0.0749269i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.64 + 0.173i)T \)
13 \( 1 + (0.197 - 3.60i)T \)
good3 \( 1 + 2.39T + 3T^{2} \)
5 \( 1 + (0.561 + 0.972i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 - 5.57T + 11T^{2} \)
17 \( 1 + (3.66 + 6.35i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 6.63T + 19T^{2} \)
23 \( 1 + (-1.69 + 2.94i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.929 - 1.60i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.60 - 4.51i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.89 + 3.28i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.55 + 2.69i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.97 - 6.89i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.44 + 2.51i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.29 - 9.16i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.35 + 4.07i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 4.95T + 61T^{2} \)
67 \( 1 - 0.510T + 67T^{2} \)
71 \( 1 + (-3.07 + 5.32i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.68 - 6.38i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.36 - 4.09i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.80T + 83T^{2} \)
89 \( 1 + (4.18 - 7.24i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.08 + 3.60i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.22688977885848807429170420602, −11.60525968324251502444478339227, −11.02873168776849178144219108776, −9.491860144173119783100285505640, −8.726001532562200799147561993942, −7.18780278202946094143957530049, −6.51701781810564834467559484251, −5.11788121434211840351206013760, −4.48020009047940889844802867918, −1.11258520739586448069622318586, 1.34542170971051231906626574394, 3.68479750323322531421322466146, 5.03802446597916962190101032192, 6.22138867202972524136349153542, 7.42008780356272839301551815768, 8.657888022630951471925542120845, 9.938677299976271073313177470231, 11.04632468529545967877909747830, 11.41045830884318948603543877770, 12.11786121537305731867597581950

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