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Properties

Label 2-180-45.4-c1-0-1
Degree $2$
Conductor $180$
Sign $0.497 - 0.867i$
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  + (0.690 + 1.58i)3-s + (1.99 + 1.00i)5-s + (−1.11 + 0.644i)7-s + (−2.04 + 2.19i)9-s + (−2.54 − 4.41i)11-s + (3.09 + 1.78i)13-s + (−0.225 + 3.86i)15-s + 0.895i·17-s + 5.34·19-s + (−1.79 − 1.32i)21-s + (−4.38 − 2.53i)23-s + (2.96 + 4.02i)25-s + (−4.89 − 1.73i)27-s + (−1.5 − 2.59i)29-s + (3.29 − 5.71i)31-s + ⋯
L(s)  = 1  + (0.398 + 0.917i)3-s + (0.892 + 0.451i)5-s + (−0.421 + 0.243i)7-s + (−0.682 + 0.731i)9-s + (−0.767 − 1.32i)11-s + (0.857 + 0.495i)13-s + (−0.0582 + 0.998i)15-s + 0.217i·17-s + 1.22·19-s + (−0.391 − 0.289i)21-s + (−0.914 − 0.528i)23-s + (0.592 + 0.805i)25-s + (−0.942 − 0.334i)27-s + (−0.278 − 0.482i)29-s + (0.592 − 1.02i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.497 - 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.497 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.497 - 0.867i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1/2),\ 0.497 - 0.867i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19860 + 0.694166i\)
\(L(\frac12)\) \(\approx\) \(1.19860 + 0.694166i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.690 - 1.58i)T \)
5 \( 1 + (-1.99 - 1.00i)T \)
good7 \( 1 + (1.11 - 0.644i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.54 + 4.41i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.09 - 1.78i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 0.895iT - 17T^{2} \)
19 \( 1 - 5.34T + 19T^{2} \)
23 \( 1 + (4.38 + 2.53i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.29 + 5.71i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.24iT - 37T^{2} \)
41 \( 1 + (-3.92 + 6.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (9.46 - 5.46i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.57 + 1.48i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 4.78iT - 53T^{2} \)
59 \( 1 + (2.87 - 4.98i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.17 + 3.75i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.39 - 4.26i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.34T + 71T^{2} \)
73 \( 1 - 9.34iT - 73T^{2} \)
79 \( 1 + (0.370 + 0.642i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.97 - 4.60i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 9.24T + 89T^{2} \)
97 \( 1 + (2.99 - 1.72i)T + (48.5 - 84.0i)T^{2} \)
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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

−13.21909835924643497078008155390, −11.52641874793360072419322086869, −10.71083421287312118586788568495, −9.832844801643587054851911828238, −8.992597543434238558488552335213, −7.951696741560846703199881544202, −6.19251733231798800694910835674, −5.47563461965773395756034294582, −3.73240743456080660069377403524, −2.58572409957873291947244887618, 1.56166982342143733085019544170, 3.09619077364357864449953345219, 5.08409454321386037953576726410, 6.25143074770011809307724219088, 7.33365894188642479876484664646, 8.344986055690163788037906983882, 9.531211437112807281535248056206, 10.25546847449157402584238127857, 11.83383854871041979532141526658, 12.71908452810682180074425736731

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