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Properties

Label 2-180-36.23-c1-0-22
Degree $2$
Conductor $180$
Sign $0.582 + 0.812i$
Analytic cond. $1.43730$
Root an. cond. $1.19887$
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  + (1.18 − 0.769i)2-s + (1.70 − 0.296i)3-s + (0.815 − 1.82i)4-s + (−0.866 − 0.5i)5-s + (1.79 − 1.66i)6-s + (−3.55 + 2.05i)7-s + (−0.438 − 2.79i)8-s + (2.82 − 1.01i)9-s + (−1.41 + 0.0733i)10-s + (1.28 + 2.23i)11-s + (0.850 − 3.35i)12-s + (−1.23 + 2.14i)13-s + (−2.63 + 5.17i)14-s + (−1.62 − 0.596i)15-s + (−2.67 − 2.97i)16-s + 5.59i·17-s + ⋯
L(s)  = 1  + (0.838 − 0.544i)2-s + (0.985 − 0.170i)3-s + (0.407 − 0.913i)4-s + (−0.387 − 0.223i)5-s + (0.733 − 0.679i)6-s + (−1.34 + 0.775i)7-s + (−0.155 − 0.987i)8-s + (0.941 − 0.336i)9-s + (−0.446 + 0.0231i)10-s + (0.388 + 0.672i)11-s + (0.245 − 0.969i)12-s + (−0.343 + 0.595i)13-s + (−0.705 + 1.38i)14-s + (−0.419 − 0.154i)15-s + (−0.667 − 0.744i)16-s + 1.35i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.582 + 0.812i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.582 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.582 + 0.812i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1/2),\ 0.582 + 0.812i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82356 - 0.936587i\)
\(L(\frac12)\) \(\approx\) \(1.82356 - 0.936587i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.18 + 0.769i)T \)
3 \( 1 + (-1.70 + 0.296i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
good7 \( 1 + (3.55 - 2.05i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.28 - 2.23i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.23 - 2.14i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 5.59iT - 17T^{2} \)
19 \( 1 + 0.255iT - 19T^{2} \)
23 \( 1 + (-3.58 + 6.21i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.78 + 2.76i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (8.20 + 4.73i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 5.63T + 37T^{2} \)
41 \( 1 + (3.64 + 2.10i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.55 - 1.47i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.96 - 3.41i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 4.80iT - 53T^{2} \)
59 \( 1 + (-0.413 + 0.717i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.47 - 4.28i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.51 + 4.33i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.80T + 71T^{2} \)
73 \( 1 + 1.18T + 73T^{2} \)
79 \( 1 + (-3.87 + 2.23i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.65 + 6.32i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 6.33iT - 89T^{2} \)
97 \( 1 + (-0.431 - 0.746i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.59593461440206300056212364193, −12.05207033663716144483387067427, −10.47107472641066848811263393025, −9.517116084553332514906622537470, −8.742103435067594189336804179961, −7.08174778118300654088193660656, −6.19917389969111783048891953375, −4.46119557333098154789094472908, −3.40252626475217045146080750251, −2.14103960537463941049452787258, 3.13035899496475119656551832123, 3.57816776397407274647010012209, 5.15194784195663393922790060776, 6.86392930824716810789732285411, 7.30761199988378525528904062350, 8.623867092882966214801972890565, 9.681162823811478410402833751491, 10.87640388440627812831895962205, 12.18257120910645697889123391082, 13.17743131162893234228953784206

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