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Properties

Label 2-180-180.43-c1-0-13
Degree $2$
Conductor $180$
Sign $0.998 + 0.0551i$
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  + (−0.0967 + 1.41i)2-s + (−1.60 − 0.658i)3-s + (−1.98 − 0.273i)4-s + (1.07 − 1.95i)5-s + (1.08 − 2.19i)6-s + (2.10 − 0.564i)7-s + (0.577 − 2.76i)8-s + (2.13 + 2.11i)9-s + (2.65 + 1.71i)10-s + (0.321 + 0.185i)11-s + (2.99 + 1.74i)12-s + (1.32 − 4.96i)13-s + (0.592 + 3.02i)14-s + (−3.01 + 2.42i)15-s + (3.85 + 1.08i)16-s + (−1.43 + 1.43i)17-s + ⋯
L(s)  = 1  + (−0.0684 + 0.997i)2-s + (−0.924 − 0.380i)3-s + (−0.990 − 0.136i)4-s + (0.482 − 0.875i)5-s + (0.442 − 0.896i)6-s + (0.795 − 0.213i)7-s + (0.204 − 0.978i)8-s + (0.710 + 0.703i)9-s + (0.840 + 0.541i)10-s + (0.0969 + 0.0559i)11-s + (0.864 + 0.503i)12-s + (0.368 − 1.37i)13-s + (0.158 + 0.808i)14-s + (−0.779 + 0.626i)15-s + (0.962 + 0.270i)16-s + (−0.348 + 0.348i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0551i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.998 + 0.0551i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1/2),\ 0.998 + 0.0551i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.895037 - 0.0247170i\)
\(L(\frac12)\) \(\approx\) \(0.895037 - 0.0247170i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0967 - 1.41i)T \)
3 \( 1 + (1.60 + 0.658i)T \)
5 \( 1 + (-1.07 + 1.95i)T \)
good7 \( 1 + (-2.10 + 0.564i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.321 - 0.185i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.32 + 4.96i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (1.43 - 1.43i)T - 17iT^{2} \)
19 \( 1 - 7.37T + 19T^{2} \)
23 \( 1 + (6.69 + 1.79i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (0.694 + 0.400i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.847 - 0.489i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.56 + 2.56i)T - 37iT^{2} \)
41 \( 1 + (-3.35 - 5.81i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.21 - 4.52i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-1.24 + 0.334i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.808 - 0.808i)T + 53iT^{2} \)
59 \( 1 + (2.13 + 3.69i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.10 - 10.5i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.79 - 10.4i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 10.1iT - 71T^{2} \)
73 \( 1 + (-3.32 - 3.32i)T + 73iT^{2} \)
79 \( 1 + (-7.04 + 12.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.68 + 6.30i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 12.7iT - 89T^{2} \)
97 \( 1 + (-1.98 - 7.40i)T + (-84.0 + 48.5i)T^{2} \)
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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

−12.86072012212159097951021253790, −11.85933074027251423882992099416, −10.54805436112695737627604611671, −9.596375742198949175629774144825, −8.209820489232666317241561972049, −7.58926676466369438860565945379, −6.06826756611730523660476870517, −5.42025382321100246687920294774, −4.39921847253209305785555740424, −1.10768816882476241131887056381, 1.82862693874564345306129668800, 3.68415684661933725239091511031, 4.95013763853685122614779505339, 6.08076202383770925505079198223, 7.50090291783759212519433723736, 9.174443887598298005517417016638, 9.856918583098167943415658686849, 10.95973289281108899218430553450, 11.50572175825099835397439922555, 12.16340675673770360745670766791

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