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Properties

Label 2-180-45.22-c2-0-8
Degree $2$
Conductor $180$
Sign $0.880 + 0.473i$
Analytic cond. $4.90464$
Root an. cond. $2.21464$
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  + (2.56 − 1.55i)3-s + (1.50 + 4.76i)5-s + (2.83 − 10.5i)7-s + (4.15 − 7.98i)9-s + (0.105 + 0.182i)11-s + (3.58 + 13.3i)13-s + (11.2 + 9.88i)15-s + (13.8 + 13.8i)17-s − 24.3i·19-s + (−9.20 − 31.5i)21-s + (−3.28 − 12.2i)23-s + (−20.4 + 14.3i)25-s + (−1.76 − 26.9i)27-s + (21.5 − 12.4i)29-s + (−25.1 + 43.5i)31-s + ⋯
L(s)  = 1  + (0.854 − 0.518i)3-s + (0.300 + 0.953i)5-s + (0.405 − 1.51i)7-s + (0.461 − 0.887i)9-s + (0.00957 + 0.0165i)11-s + (0.275 + 1.02i)13-s + (0.751 + 0.659i)15-s + (0.812 + 0.812i)17-s − 1.27i·19-s + (−0.438 − 1.50i)21-s + (−0.142 − 0.532i)23-s + (−0.818 + 0.573i)25-s + (−0.0655 − 0.997i)27-s + (0.744 − 0.430i)29-s + (−0.810 + 1.40i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.880 + 0.473i$
Analytic conductor: \(4.90464\)
Root analytic conductor: \(2.21464\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1),\ 0.880 + 0.473i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.06372 - 0.519695i\)
\(L(\frac12)\) \(\approx\) \(2.06372 - 0.519695i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-2.56 + 1.55i)T \)
5 \( 1 + (-1.50 - 4.76i)T \)
good7 \( 1 + (-2.83 + 10.5i)T + (-42.4 - 24.5i)T^{2} \)
11 \( 1 + (-0.105 - 0.182i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-3.58 - 13.3i)T + (-146. + 84.5i)T^{2} \)
17 \( 1 + (-13.8 - 13.8i)T + 289iT^{2} \)
19 \( 1 + 24.3iT - 361T^{2} \)
23 \( 1 + (3.28 + 12.2i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 + (-21.5 + 12.4i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (25.1 - 43.5i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (18.2 + 18.2i)T + 1.36e3iT^{2} \)
41 \( 1 + (13.8 - 24.0i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (14.4 + 3.87i)T + (1.60e3 + 924.5i)T^{2} \)
47 \( 1 + (20.7 - 77.4i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (57.6 - 57.6i)T - 2.80e3iT^{2} \)
59 \( 1 + (21.3 + 12.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (14.2 + 24.7i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (42.7 - 11.4i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 118.T + 5.04e3T^{2} \)
73 \( 1 + (53.2 - 53.2i)T - 5.32e3iT^{2} \)
79 \( 1 + (-18.1 + 10.4i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (29.6 + 7.95i)T + (5.96e3 + 3.44e3i)T^{2} \)
89 \( 1 - 39.1iT - 7.92e3T^{2} \)
97 \( 1 + (-16.4 + 61.5i)T + (-8.14e3 - 4.70e3i)T^{2} \)
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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

−12.53879188880769312321682354191, −11.17729344826709193740626979210, −10.41416216077863031773229180821, −9.352293490646469787536993968532, −8.080292302217861513138004912866, −7.12667677465759140760137498997, −6.48333165695329883460380810835, −4.34313004538630974443158563427, −3.17116179316596894325415743773, −1.52640766601350778020805604848, 1.90171157087229423052301092089, 3.38276714671768227590294097024, 5.05796340797376152901134360612, 5.70124151223980454305656783873, 7.87754327456726461031253866859, 8.480676788962097835131324934266, 9.392571846976796117010131622996, 10.19084698745944380510966305720, 11.72446548519618735548138025904, 12.52530032600545481491830791760

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