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Properties

Label 2-182-91.82-c2-0-8
Degree $2$
Conductor $182$
Sign $0.964 - 0.265i$
Analytic cond. $4.95914$
Root an. cond. $2.22691$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 + 0.707i)2-s + 3.15i·3-s + (0.999 − 1.73i)4-s + (2.18 − 3.78i)5-s + (−2.23 − 3.86i)6-s + (5.93 − 3.71i)7-s + 2.82i·8-s − 0.964·9-s + 6.17i·10-s − 15.8i·11-s + (5.46 + 3.15i)12-s + (−4.43 − 12.2i)13-s + (−4.64 + 8.74i)14-s + (11.9 + 6.89i)15-s + (−2.00 − 3.46i)16-s + (27.1 + 15.6i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + 1.05i·3-s + (0.249 − 0.433i)4-s + (0.436 − 0.756i)5-s + (−0.372 − 0.644i)6-s + (0.847 − 0.530i)7-s + 0.353i·8-s − 0.107·9-s + 0.617i·10-s − 1.44i·11-s + (0.455 + 0.263i)12-s + (−0.340 − 0.940i)13-s + (−0.331 + 0.624i)14-s + (0.795 + 0.459i)15-s + (−0.125 − 0.216i)16-s + (1.59 + 0.922i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(182\)    =    \(2 \cdot 7 \cdot 13\)
Sign: $0.964 - 0.265i$
Analytic conductor: \(4.95914\)
Root analytic conductor: \(2.22691\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{182} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 182,\ (\ :1),\ 0.964 - 0.265i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.32007 + 0.178694i\)
\(L(\frac12)\) \(\approx\) \(1.32007 + 0.178694i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 - 0.707i)T \)
7 \( 1 + (-5.93 + 3.71i)T \)
13 \( 1 + (4.43 + 12.2i)T \)
good3 \( 1 - 3.15iT - 9T^{2} \)
5 \( 1 + (-2.18 + 3.78i)T + (-12.5 - 21.6i)T^{2} \)
11 \( 1 + 15.8iT - 121T^{2} \)
17 \( 1 + (-27.1 - 15.6i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + 14.5T + 361T^{2} \)
23 \( 1 + (-7.40 - 12.8i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (0.175 - 0.304i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (1.96 + 3.40i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (4.24 - 2.44i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-33.1 + 57.4i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (24.7 + 42.9i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (42.0 - 72.7i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-9.86 - 17.0i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (34.9 - 60.6i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 - 14.7iT - 3.72e3T^{2} \)
67 \( 1 + 116. iT - 4.48e3T^{2} \)
71 \( 1 + (-9.64 + 5.56i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (4.36 + 7.56i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (55.6 - 96.3i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 140.T + 6.88e3T^{2} \)
89 \( 1 + (-57.5 - 99.7i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (5.49 + 9.52i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−12.40727228689231755334338990303, −10.89406603178818525843069129028, −10.48978450198810685499606640674, −9.421406182786219863039304530442, −8.485405925185420424209880705120, −7.65349803586409647815279788146, −5.80240447781985230719314829290, −5.06885745752111661364100330311, −3.61062381713325729645647282302, −1.15621916337684214638239892362, 1.61924998060201331553793882448, 2.53527112543706937844817367099, 4.74451569852455989381686929327, 6.49216862712457557688753734473, 7.25145893661840522820179395658, 8.114322590568527899201483891498, 9.504118347665308260910302870147, 10.27198772116579806280368299842, 11.59845823946232325583441087965, 12.17478381741298765244685280335

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