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Properties

Label 2-180-180.59-c1-0-0
Degree $2$
Conductor $180$
Sign $-0.134 - 0.990i$
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  + (−0.937 + 1.05i)2-s + (−0.959 − 1.44i)3-s + (−0.242 − 1.98i)4-s + (−1.62 + 1.53i)5-s + (2.42 + 0.335i)6-s + (0.550 + 0.953i)7-s + (2.32 + 1.60i)8-s + (−1.15 + 2.76i)9-s + (−0.106 − 3.16i)10-s + (2.84 + 4.92i)11-s + (−2.62 + 2.25i)12-s + (2.07 + 1.19i)13-s + (−1.52 − 0.310i)14-s + (3.77 + 0.864i)15-s + (−3.88 + 0.963i)16-s − 1.29·17-s + ⋯
L(s)  = 1  + (−0.662 + 0.748i)2-s + (−0.554 − 0.832i)3-s + (−0.121 − 0.992i)4-s + (−0.725 + 0.687i)5-s + (0.990 + 0.136i)6-s + (0.208 + 0.360i)7-s + (0.823 + 0.567i)8-s + (−0.385 + 0.922i)9-s + (−0.0337 − 0.999i)10-s + (0.856 + 1.48i)11-s + (−0.759 + 0.651i)12-s + (0.575 + 0.332i)13-s + (−0.407 − 0.0830i)14-s + (0.974 + 0.223i)15-s + (−0.970 + 0.240i)16-s − 0.313·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.134 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.134 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.134 - 0.990i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1/2),\ -0.134 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.369268 + 0.422729i\)
\(L(\frac12)\) \(\approx\) \(0.369268 + 0.422729i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.937 - 1.05i)T \)
3 \( 1 + (0.959 + 1.44i)T \)
5 \( 1 + (1.62 - 1.53i)T \)
good7 \( 1 + (-0.550 - 0.953i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.84 - 4.92i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.07 - 1.19i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.29T + 17T^{2} \)
19 \( 1 - 2.78iT - 19T^{2} \)
23 \( 1 + (1.82 + 1.05i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.51 - 3.18i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.78 + 1.60i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.51iT - 37T^{2} \)
41 \( 1 + (-6.35 - 3.66i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.25 - 3.90i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-8.09 + 4.67i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 11.5T + 53T^{2} \)
59 \( 1 + (-3.66 + 6.34i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.48 - 4.31i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.85 + 3.20i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.6T + 71T^{2} \)
73 \( 1 - 9.86iT - 73T^{2} \)
79 \( 1 + (-0.975 + 0.563i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (9.98 - 5.76i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 9.16iT - 89T^{2} \)
97 \( 1 + (-11.1 + 6.42i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.81701808726772404494098866113, −11.76094597055368052161037550729, −11.05771154138261029761402017845, −9.897749896630085445872916699499, −8.615178190051327080216184570733, −7.55314671465298273332149839707, −6.87999160174482884131794889501, −5.94481097110200984688991436411, −4.41691809983620885526994138106, −1.83582070134672532771372003979, 0.71949135177582918393029392960, 3.52641069955107743623448972632, 4.27228134113748501529547769373, 5.85797111591006112098886185638, 7.54209809732547756402059296883, 8.809242436225438925120892039576, 9.208750045002700673613344282173, 10.78259174531008172035975639234, 11.16819156474263256066955719577, 11.98484979721057761017041682917

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