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Properties

Label 2-182-91.10-c2-0-8
Degree $2$
Conductor $182$
Sign $-0.642 - 0.766i$
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 + 4.02i·3-s + (0.999 + 1.73i)4-s + (4.59 + 7.96i)5-s + (−2.84 + 4.93i)6-s + (−5.39 − 4.45i)7-s + 2.82i·8-s − 7.23·9-s + 13.0i·10-s − 17.1i·11-s + (−6.97 + 4.02i)12-s + (9.51 − 8.85i)13-s + (−3.46 − 9.27i)14-s + (−32.0 + 18.5i)15-s + (−2.00 + 3.46i)16-s + (3.18 − 1.83i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + 1.34i·3-s + (0.249 + 0.433i)4-s + (0.919 + 1.59i)5-s + (−0.474 + 0.822i)6-s + (−0.771 − 0.636i)7-s + 0.353i·8-s − 0.803·9-s + 1.30i·10-s − 1.56i·11-s + (−0.581 + 0.335i)12-s + (0.731 − 0.681i)13-s + (−0.247 − 0.662i)14-s + (−2.13 + 1.23i)15-s + (−0.125 + 0.216i)16-s + (0.187 − 0.108i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(182\)    =    \(2 \cdot 7 \cdot 13\)
Sign: $-0.642 - 0.766i$
Analytic conductor: \(4.95914\)
Root analytic conductor: \(2.22691\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{182} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 182,\ (\ :1),\ -0.642 - 0.766i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.955199 + 2.04612i\)
\(L(\frac12)\) \(\approx\) \(0.955199 + 2.04612i\)
\(L(2)\) 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 \( 1 + (5.39 + 4.45i)T \)
13 \( 1 + (-9.51 + 8.85i)T \)
good3 \( 1 - 4.02iT - 9T^{2} \)
5 \( 1 + (-4.59 - 7.96i)T + (-12.5 + 21.6i)T^{2} \)
11 \( 1 + 17.1iT - 121T^{2} \)
17 \( 1 + (-3.18 + 1.83i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 - 17.2T + 361T^{2} \)
23 \( 1 + (3.50 - 6.06i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-0.359 - 0.623i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (22.2 - 38.4i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-26.4 - 15.2i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (16.2 + 28.1i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-26.8 + 46.4i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (36.6 + 63.5i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (4.39 - 7.62i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (6.58 + 11.4i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 - 11.6iT - 3.72e3T^{2} \)
67 \( 1 + 27.7iT - 4.48e3T^{2} \)
71 \( 1 + (25.6 + 14.7i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (30.1 - 52.1i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (66.3 + 114. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 108.T + 6.88e3T^{2} \)
89 \( 1 + (-48.6 + 84.2i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-39.5 + 68.5i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−13.26306514178071269586454063631, −11.40715161821027659722073022557, −10.58904183294868373898242444973, −10.14952230836060954263207581163, −8.957216768774234566870326368862, −7.30054426167968013568624582682, −6.19096050940188204403581575337, −5.44831016058329197444665695078, −3.42683101693864448644433377100, −3.30399843619337399645263370641, 1.29914256771974732590517414297, 2.25340215411286514550859949257, 4.46823823905437262286308240755, 5.71830025777063944535852063185, 6.50278615575196987433036858277, 7.84487656937696425651513819008, 9.260487470417592573495067082334, 9.744868041222793161386386317805, 11.68897691165237307380963079792, 12.43846026043508612635075147635

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