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Properties

Label 2-182-13.10-c1-0-5
Degree $2$
Conductor $182$
Sign $0.265 + 0.964i$
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 + (−0.366 − 0.633i)3-s + (0.499 − 0.866i)4-s − i·5-s + (−0.633 − 0.366i)6-s + (−0.866 − 0.5i)7-s − 0.999i·8-s + (1.23 − 2.13i)9-s + (−0.5 − 0.866i)10-s + (0.633 − 0.366i)11-s − 0.732·12-s + (2.59 + 2.5i)13-s − 0.999·14-s + (−0.633 + 0.366i)15-s + (−0.5 − 0.866i)16-s + (−2.86 + 4.96i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.211 − 0.366i)3-s + (0.249 − 0.433i)4-s − 0.447i·5-s + (−0.258 − 0.149i)6-s + (−0.327 − 0.188i)7-s − 0.353i·8-s + (0.410 − 0.711i)9-s + (−0.158 − 0.273i)10-s + (0.191 − 0.110i)11-s − 0.211·12-s + (0.720 + 0.693i)13-s − 0.267·14-s + (−0.163 + 0.0945i)15-s + (−0.125 − 0.216i)16-s + (−0.695 + 1.20i)17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(182\)    =    \(2 \cdot 7 \cdot 13\)
Sign: $0.265 + 0.964i$
Analytic conductor: \(1.45327\)
Root analytic conductor: \(1.20551\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{182} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 182,\ (\ :1/2),\ 0.265 + 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19410 - 0.910203i\)
\(L(\frac12)\) \(\approx\) \(1.19410 - 0.910203i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.59 - 2.5i)T \)
good3 \( 1 + (0.366 + 0.633i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + iT - 5T^{2} \)
11 \( 1 + (-0.633 + 0.366i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.86 - 4.96i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.26 + 0.732i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.633 - 1.09i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.26iT - 31T^{2} \)
37 \( 1 + (-4.5 + 2.59i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.13 - 1.23i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.09 - 10.5i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.92iT - 47T^{2} \)
53 \( 1 - 1.53T + 53T^{2} \)
59 \( 1 + (9.29 + 5.36i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.86 + 10.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.0 + 5.83i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (12 + 6.92i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 11.3iT - 73T^{2} \)
79 \( 1 + 3.80T + 79T^{2} \)
83 \( 1 + 3.80iT - 83T^{2} \)
89 \( 1 + (-2.19 + 1.26i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.73 + 2.73i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.71669137373022371456803013183, −11.59157489655295214453648062840, −10.71810185023890674291524276066, −9.484990897303523392060034220056, −8.496242160014646119182298919834, −6.80875986401046540840311704036, −6.20627802977341706734878540628, −4.63491214927401269796912183972, −3.52754363132540193608612426474, −1.46261959485090427232875407989, 2.68580633735142863466309283617, 4.16384547553625417893385857857, 5.31133512543390713713315983042, 6.47635299543540557208767977245, 7.48523632906071525062112076402, 8.732153072542549358022402804286, 10.05718927290876612957994908218, 10.93987794058558637411367441395, 11.85980151036293406223503862465, 13.12117384810670646995145093251

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