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Properties

Label 2-182-13.12-c1-0-3
Degree $2$
Conductor $182$
Sign $0.502 + 0.864i$
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  + i·2-s − 3.30·3-s − 4-s + 0.376i·5-s − 3.30i·6-s − i·7-s − i·8-s + 7.92·9-s − 0.376·10-s − 5.68i·11-s + 3.30·12-s + (−1.81 − 3.11i)13-s + 14-s − 1.24i·15-s + 16-s − 1.24·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.90·3-s − 0.5·4-s + 0.168i·5-s − 1.34i·6-s − 0.377i·7-s − 0.353i·8-s + 2.64·9-s − 0.119·10-s − 1.71i·11-s + 0.954·12-s + (−0.502 − 0.864i)13-s + 0.267·14-s − 0.321i·15-s + 0.250·16-s − 0.302·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.502 + 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.502 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(182\)    =    \(2 \cdot 7 \cdot 13\)
Sign: $0.502 + 0.864i$
Analytic conductor: \(1.45327\)
Root analytic conductor: \(1.20551\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{182} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 182,\ (\ :1/2),\ 0.502 + 0.864i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.358952 - 0.206570i\)
\(L(\frac12)\) \(\approx\) \(0.358952 - 0.206570i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
7 \( 1 + iT \)
13 \( 1 + (1.81 + 3.11i)T \)
good3 \( 1 + 3.30T + 3T^{2} \)
5 \( 1 - 0.376iT - 5T^{2} \)
11 \( 1 + 5.68iT - 11T^{2} \)
17 \( 1 + 1.24T + 17T^{2} \)
19 \( 1 - 1.62iT - 19T^{2} \)
23 \( 1 + 4.92T + 23T^{2} \)
29 \( 1 + 4.61T + 29T^{2} \)
31 \( 1 + 5.68iT - 31T^{2} \)
37 \( 1 + 6.92iT - 37T^{2} \)
41 \( 1 + 1.07iT - 41T^{2} \)
43 \( 1 - 3.24T + 43T^{2} \)
47 \( 1 + 2.31iT - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 11.4iT - 59T^{2} \)
61 \( 1 - 5.30T + 61T^{2} \)
67 \( 1 - 8.92iT - 67T^{2} \)
71 \( 1 - 13.2iT - 71T^{2} \)
73 \( 1 - 0.317iT - 73T^{2} \)
79 \( 1 - 2.17T + 79T^{2} \)
83 \( 1 - 9.74iT - 83T^{2} \)
89 \( 1 - 9.24iT - 89T^{2} \)
97 \( 1 + 10.9iT - 97T^{2} \)
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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

−12.49355202758156126732306556395, −11.30590096596514879031766741900, −10.73882907397690891767619327772, −9.720349362667467102476843047774, −8.082799439667214975215450517845, −6.98439109258110554584549561599, −5.95069605969590117723883622220, −5.40075158968752373583365387633, −3.97414750535865299710012834716, −0.48312153935958298843761971831, 1.78688339710042113122248251601, 4.44158423768658849348999576109, 5.00254300325138673382042963811, 6.39132692428128971315530933615, 7.34942995264402515535720368348, 9.293151165954934610606631899122, 10.09421663504547360106066958055, 10.97697950547371886337013627397, 12.03424670442074824598343082971, 12.25010127557525646370926777039

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