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Properties

Label 2-182-91.88-c1-0-8
Degree $2$
Conductor $182$
Sign $0.775 + 0.631i$
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 + 1.92·3-s + (0.499 + 0.866i)4-s + (1.94 − 1.12i)5-s + (−1.66 − 0.961i)6-s + (−1.27 − 2.31i)7-s − 0.999i·8-s + 0.694·9-s − 2.25·10-s + 2.84i·11-s + (0.961 + 1.66i)12-s + (1.60 + 3.23i)13-s + (−0.0550 + 2.64i)14-s + (3.74 − 2.16i)15-s + (−0.5 + 0.866i)16-s + (−0.387 − 0.670i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + 1.10·3-s + (0.249 + 0.433i)4-s + (0.871 − 0.503i)5-s + (−0.679 − 0.392i)6-s + (−0.481 − 0.876i)7-s − 0.353i·8-s + 0.231·9-s − 0.711·10-s + 0.858i·11-s + (0.277 + 0.480i)12-s + (0.444 + 0.895i)13-s + (−0.0147 + 0.706i)14-s + (0.967 − 0.558i)15-s + (−0.125 + 0.216i)16-s + (−0.0939 − 0.162i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(182\)    =    \(2 \cdot 7 \cdot 13\)
Sign: $0.775 + 0.631i$
Analytic conductor: \(1.45327\)
Root analytic conductor: \(1.20551\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{182} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 182,\ (\ :1/2),\ 0.775 + 0.631i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22080 - 0.434626i\)
\(L(\frac12)\) \(\approx\) \(1.22080 - 0.434626i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.27 + 2.31i)T \)
13 \( 1 + (-1.60 - 3.23i)T \)
good3 \( 1 - 1.92T + 3T^{2} \)
5 \( 1 + (-1.94 + 1.12i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 - 2.84iT - 11T^{2} \)
17 \( 1 + (0.387 + 0.670i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 8.09iT - 19T^{2} \)
23 \( 1 + (2.31 - 4.00i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.65 - 4.60i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-6.29 - 3.63i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.63 + 2.67i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.828 - 0.478i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.76 + 3.05i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (10.2 - 5.94i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.47 - 6.02i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.02 - 2.32i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 5.02T + 61T^{2} \)
67 \( 1 + 11.2iT - 67T^{2} \)
71 \( 1 + (3.66 + 2.11i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.00 - 1.73i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.59 + 11.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.23iT - 83T^{2} \)
89 \( 1 + (-8.90 - 5.14i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.284 - 0.164i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.73741457307817444211208426236, −11.41541326583255226001578921662, −10.21400721940738776507560664783, −9.318904628630750949063917224554, −8.915153001330048704237768408854, −7.55070604250147748800646283179, −6.58449824408977295463185476261, −4.64147066328541293178440051387, −3.15273098230619654234178572152, −1.73765845155273037517404751968, 2.23959375280918858548593563615, 3.33201266736517185713503673911, 5.79211620016335296767129971210, 6.32056848776487214371150651469, 8.262699298928862364621666098461, 8.353532887322116122818105041943, 9.776475120112640987058478330336, 10.22610116556587198469584804369, 11.68794986489570102130026933006, 13.01101498550543006468514476081

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