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Properties

Label 2-182-13.4-c1-0-4
Degree $2$
Conductor $182$
Sign $0.999 - 0.0302i$
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  + (0.866 + 0.5i)2-s + (1.27 − 2.21i)3-s + (0.499 + 0.866i)4-s + 3.48i·5-s + (2.21 − 1.27i)6-s + (0.866 − 0.5i)7-s + 0.999i·8-s + (−1.76 − 3.06i)9-s + (−1.74 + 3.02i)10-s + (−2.32 − 1.34i)11-s + 2.55·12-s + (−3.15 − 1.74i)13-s + 0.999·14-s + (7.72 + 4.45i)15-s + (−0.5 + 0.866i)16-s + (−2.95 − 5.12i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.737 − 1.27i)3-s + (0.249 + 0.433i)4-s + 1.55i·5-s + (0.903 − 0.521i)6-s + (0.327 − 0.188i)7-s + 0.353i·8-s + (−0.589 − 1.02i)9-s + (−0.551 + 0.955i)10-s + (−0.700 − 0.404i)11-s + 0.737·12-s + (−0.874 − 0.484i)13-s + 0.267·14-s + (1.99 + 1.15i)15-s + (−0.125 + 0.216i)16-s + (−0.717 − 1.24i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0302i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0302i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(182\)    =    \(2 \cdot 7 \cdot 13\)
Sign: $0.999 - 0.0302i$
Analytic conductor: \(1.45327\)
Root analytic conductor: \(1.20551\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{182} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 182,\ (\ :1/2),\ 0.999 - 0.0302i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.88999 + 0.0285581i\)
\(L(\frac12)\) \(\approx\) \(1.88999 + 0.0285581i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (3.15 + 1.74i)T \)
good3 \( 1 + (-1.27 + 2.21i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 3.48iT - 5T^{2} \)
11 \( 1 + (2.32 + 1.34i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.95 + 5.12i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.50 + 2.59i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.52 - 6.10i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.56 - 6.17i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.782iT - 31T^{2} \)
37 \( 1 + (-6.76 - 3.90i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.136 - 0.0788i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.165 + 0.285i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 1.60iT - 47T^{2} \)
53 \( 1 - 3.92T + 53T^{2} \)
59 \( 1 + (-8.26 + 4.77i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.70 + 13.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.837 - 0.483i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.62 + 2.09i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 15.0iT - 73T^{2} \)
79 \( 1 + 0.293T + 79T^{2} \)
83 \( 1 - 2.87iT - 83T^{2} \)
89 \( 1 + (-4.51 - 2.60i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.73 - 1.57i)T + (48.5 - 84.0i)T^{2} \)
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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

−13.01845911871312460410245002107, −11.75215792629648012775355976644, −11.01190428038268388256619546301, −9.594761863383694122039629423499, −7.939620684624277541452409854772, −7.37308924338935735683133882310, −6.74483029828408239824074871312, −5.30656507986959897302662149245, −3.20489961173158244241654425177, −2.44368345437878058929245038739, 2.24504697733594351962728759432, 4.11411996887598604101317731594, 4.62291352703247328547663295894, 5.69138109754339902930830162996, 7.88074857799633799453515657113, 8.825144367552302686129270515044, 9.677474042990203588594064537265, 10.46126177889083490688971631944, 11.85492005158326550334182577311, 12.64554231578161366365790235240

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