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Properties

Label 2-180-60.23-c2-0-19
Degree 22
Conductor 180180
Sign −0.179+0.983i-0.179 + 0.983i
Analytic cond. 4.904644.90464
Root an. cond. 2.214642.21464
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.537 + 1.92i)2-s + (−3.42 + 2.06i)4-s + (−2.15 − 4.51i)5-s + (−4.16 − 4.16i)7-s + (−5.82 − 5.48i)8-s + (7.54 − 6.56i)10-s − 21.7·11-s + (3.43 + 3.43i)13-s + (5.78 − 10.2i)14-s + (7.43 − 14.1i)16-s + (−13.0 − 13.0i)17-s + 19.7·19-s + (16.7 + 11.0i)20-s + (−11.6 − 41.9i)22-s + (17.1 + 17.1i)23-s + ⋯
L(s)  = 1  + (0.268 + 0.963i)2-s + (−0.855 + 0.517i)4-s + (−0.430 − 0.902i)5-s + (−0.594 − 0.594i)7-s + (−0.728 − 0.685i)8-s + (0.754 − 0.656i)10-s − 1.97·11-s + (0.264 + 0.264i)13-s + (0.413 − 0.732i)14-s + (0.464 − 0.885i)16-s + (−0.767 − 0.767i)17-s + 1.03·19-s + (0.835 + 0.550i)20-s + (−0.531 − 1.90i)22-s + (0.743 + 0.743i)23-s + ⋯

Functional equation

Λ(s)=(180s/2 ΓC(s) L(s)=((−0.179+0.983i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.179 + 0.983i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(180s/2 ΓC(s+1) L(s)=((−0.179+0.983i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.179 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 180180    =    22⋅32⋅52^{2} \cdot 3^{2} \cdot 5
Sign: −0.179+0.983i-0.179 + 0.983i
Analytic conductor: 4.904644.90464
Root analytic conductor: 2.214642.21464
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ180(143,⋅)\chi_{180} (143, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 180, ( :1), −0.179+0.983i)(2,\ 180,\ (\ :1),\ -0.179 + 0.983i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.241700−0.289886i0.241700 - 0.289886i
L(12)L(\frac12) ≈\approx 0.241700−0.289886i0.241700 - 0.289886i
L(2)L(2) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(−0.537−1.92i)T 1 + (-0.537 - 1.92i)T
3 1 1
5 1+(2.15+4.51i)T 1 + (2.15 + 4.51i)T
good7 1+(4.16+4.16i)T+49iT2 1 + (4.16 + 4.16i)T + 49iT^{2}
11 1+21.7T+121T2 1 + 21.7T + 121T^{2}
13 1+(−3.43−3.43i)T+169iT2 1 + (-3.43 - 3.43i)T + 169iT^{2}
17 1+(13.0+13.0i)T+289iT2 1 + (13.0 + 13.0i)T + 289iT^{2}
19 1−19.7T+361T2 1 - 19.7T + 361T^{2}
23 1+(−17.1−17.1i)T+529iT2 1 + (-17.1 - 17.1i)T + 529iT^{2}
29 1+20.8T+841T2 1 + 20.8T + 841T^{2}
31 1+12.9iT−961T2 1 + 12.9iT - 961T^{2}
37 1+(21.0−21.0i)T−1.36e3iT2 1 + (21.0 - 21.0i)T - 1.36e3iT^{2}
41 1+58.0iT−1.68e3T2 1 + 58.0iT - 1.68e3T^{2}
43 1+(11.2−11.2i)T−1.84e3iT2 1 + (11.2 - 11.2i)T - 1.84e3iT^{2}
47 1+(4.90−4.90i)T−2.20e3iT2 1 + (4.90 - 4.90i)T - 2.20e3iT^{2}
53 1+(52.7−52.7i)T−2.80e3iT2 1 + (52.7 - 52.7i)T - 2.80e3iT^{2}
59 1+72.9iT−3.48e3T2 1 + 72.9iT - 3.48e3T^{2}
61 1−2.48T+3.72e3T2 1 - 2.48T + 3.72e3T^{2}
67 1+(0.141+0.141i)T+4.48e3iT2 1 + (0.141 + 0.141i)T + 4.48e3iT^{2}
71 1+16.7T+5.04e3T2 1 + 16.7T + 5.04e3T^{2}
73 1+(28.6+28.6i)T+5.32e3iT2 1 + (28.6 + 28.6i)T + 5.32e3iT^{2}
79 1+111.T+6.24e3T2 1 + 111.T + 6.24e3T^{2}
83 1+(−99.0−99.0i)T+6.88e3iT2 1 + (-99.0 - 99.0i)T + 6.88e3iT^{2}
89 1−47.0T+7.92e3T2 1 - 47.0T + 7.92e3T^{2}
97 1+(−49.4+49.4i)T−9.40e3iT2 1 + (-49.4 + 49.4i)T - 9.40e3iT^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−12.57087556024491556034162685994, −11.28069342435312403711526673005, −9.867495804550926127182484447306, −8.936996629804878776884124520745, −7.80074920730372764254656495777, −7.13975761949688160584443210812, −5.54396726525615958697525819951, −4.74913188798492732260555287755, −3.33174738619211621137901158304, −0.19690213006866818465244202301, 2.50913372303506341699906715921, 3.35808986507978360796935762522, 5.00794389846791223912161927269, 6.16326408063781655359540922555, 7.69469736108558190458548125754, 8.829602242815632390982021205982, 10.13820937634531936201628681143, 10.71946275133329164601257329613, 11.61392111070846241019114411938, 12.81993271685713434011326837465

Graph of the ZZ-function along the critical line