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Properties

Label 2-182-91.9-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: $\chi_{182} (9, \cdot )$
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

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

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