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Properties

Label 2-180-180.103-c1-0-17
Degree $2$
Conductor $180$
Sign $0.925 + 0.378i$
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  + (−0.0749 + 1.41i)2-s + (−1.68 − 0.420i)3-s + (−1.98 − 0.211i)4-s + (0.993 − 2.00i)5-s + (0.719 − 2.34i)6-s + (0.596 − 2.22i)7-s + (0.447 − 2.79i)8-s + (2.64 + 1.41i)9-s + (2.75 + 1.55i)10-s + (2.26 − 1.30i)11-s + (3.25 + 1.19i)12-s + (−3.71 + 0.996i)13-s + (3.09 + 1.00i)14-s + (−2.51 + 2.94i)15-s + (3.91 + 0.841i)16-s + (3.98 − 3.98i)17-s + ⋯
L(s)  = 1  + (−0.0529 + 0.998i)2-s + (−0.970 − 0.242i)3-s + (−0.994 − 0.105i)4-s + (0.444 − 0.895i)5-s + (0.293 − 0.955i)6-s + (0.225 − 0.841i)7-s + (0.158 − 0.987i)8-s + (0.882 + 0.470i)9-s + (0.871 + 0.491i)10-s + (0.683 − 0.394i)11-s + (0.939 + 0.343i)12-s + (−1.03 + 0.276i)13-s + (0.827 + 0.269i)14-s + (−0.648 + 0.761i)15-s + (0.977 + 0.210i)16-s + (0.965 − 0.965i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.378i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.925 + 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.925 + 0.378i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1/2),\ 0.925 + 0.378i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.782810 - 0.153833i\)
\(L(\frac12)\) \(\approx\) \(0.782810 - 0.153833i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0749 - 1.41i)T \)
3 \( 1 + (1.68 + 0.420i)T \)
5 \( 1 + (-0.993 + 2.00i)T \)
good7 \( 1 + (-0.596 + 2.22i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-2.26 + 1.30i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.71 - 0.996i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-3.98 + 3.98i)T - 17iT^{2} \)
19 \( 1 + 1.39T + 19T^{2} \)
23 \( 1 + (1.42 + 5.30i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-1.10 + 0.635i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.32 - 3.07i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.77 - 6.77i)T - 37iT^{2} \)
41 \( 1 + (-0.436 + 0.756i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.09 - 0.294i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (1.24 - 4.65i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-5.13 - 5.13i)T + 53iT^{2} \)
59 \( 1 + (-1.28 + 2.22i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.99 + 8.65i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.08 - 1.09i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 16.0iT - 71T^{2} \)
73 \( 1 + (-3.33 - 3.33i)T + 73iT^{2} \)
79 \( 1 + (-6.00 - 10.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-14.6 - 3.91i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + 11.6iT - 89T^{2} \)
97 \( 1 + (-10.5 - 2.81i)T + (84.0 + 48.5i)T^{2} \)
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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

−12.58376963085644424548277242451, −11.92953682742305504460398199804, −10.35923323399146678311016392558, −9.613273799894951124413710322315, −8.333790954112133315276357671219, −7.22303012070902922026017733681, −6.31561891208427840475246011397, −5.11814137407905428499469162736, −4.39293345413256570281136788636, −0.916908811390773034396028530879, 1.98186689339857205357679202413, 3.65977868883752779557817852941, 5.15138169714863451721320589842, 6.08442002050523472805647946959, 7.57169989841704993986949175925, 9.196135926337636852931641742903, 10.05848361710289362503333259619, 10.68089349300123496634087672642, 12.01593049847979222140987107259, 12.06469736517967706632382434499

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