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Properties

Label 2-360-120.59-c3-0-1
Degree $2$
Conductor $360$
Sign $-0.985 - 0.169i$
Analytic cond. $21.2406$
Root an. cond. $4.60876$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.51 + 2.38i)2-s + (−3.38 − 7.24i)4-s + (−6.07 − 9.38i)5-s − 14.2·7-s + (22.4 + 2.93i)8-s + (31.6 − 0.252i)10-s − 27.1i·11-s + 26.3·13-s + (21.6 − 34.0i)14-s + (−41.0 + 49.0i)16-s + 85.4·17-s − 42.9·19-s + (−47.4 + 75.8i)20-s + (64.7 + 41.2i)22-s − 192. i·23-s + ⋯
L(s)  = 1  + (−0.537 + 0.843i)2-s + (−0.423 − 0.906i)4-s + (−0.543 − 0.839i)5-s − 0.769·7-s + (0.991 + 0.129i)8-s + (0.999 − 0.00799i)10-s − 0.743i·11-s + 0.562·13-s + (0.413 − 0.649i)14-s + (−0.641 + 0.766i)16-s + 1.21·17-s − 0.519·19-s + (−0.530 + 0.847i)20-s + (0.627 + 0.399i)22-s − 1.74i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.985 - 0.169i$
Analytic conductor: \(21.2406\)
Root analytic conductor: \(4.60876\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :3/2),\ -0.985 - 0.169i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1211320423\)
\(L(\frac12)\) \(\approx\) \(0.1211320423\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.51 - 2.38i)T \)
3 \( 1 \)
5 \( 1 + (6.07 + 9.38i)T \)
good7 \( 1 + 14.2T + 343T^{2} \)
11 \( 1 + 27.1iT - 1.33e3T^{2} \)
13 \( 1 - 26.3T + 2.19e3T^{2} \)
17 \( 1 - 85.4T + 4.91e3T^{2} \)
19 \( 1 + 42.9T + 6.85e3T^{2} \)
23 \( 1 + 192. iT - 1.21e4T^{2} \)
29 \( 1 + 237.T + 2.43e4T^{2} \)
31 \( 1 - 232. iT - 2.97e4T^{2} \)
37 \( 1 + 214.T + 5.06e4T^{2} \)
41 \( 1 - 427. iT - 6.89e4T^{2} \)
43 \( 1 - 317. iT - 7.95e4T^{2} \)
47 \( 1 - 78.8iT - 1.03e5T^{2} \)
53 \( 1 - 49.0iT - 1.48e5T^{2} \)
59 \( 1 + 386. iT - 2.05e5T^{2} \)
61 \( 1 + 11.0iT - 2.26e5T^{2} \)
67 \( 1 + 275. iT - 3.00e5T^{2} \)
71 \( 1 + 965.T + 3.57e5T^{2} \)
73 \( 1 - 816. iT - 3.89e5T^{2} \)
79 \( 1 + 332. iT - 4.93e5T^{2} \)
83 \( 1 - 459.T + 5.71e5T^{2} \)
89 \( 1 - 1.05e3iT - 7.04e5T^{2} \)
97 \( 1 - 329. iT - 9.12e5T^{2} \)
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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

−11.27374323802466149212436908602, −10.33818542552918790031661185939, −9.347409936981285019798535157057, −8.549944261348009783374652342411, −7.901882505558862426321819628606, −6.67297136335096481946320539070, −5.81676067798331698028564128035, −4.72817619795435150344774734469, −3.44547650305374593974448329852, −1.18591200438850044684754446665, 0.05731222457427330040061502953, 1.91357535868447911299247236179, 3.33611621866278633501106903515, 3.92431827529709878153327054392, 5.70545930006656624449093897998, 7.17022952202163683881256458364, 7.66129135398172835083291198828, 8.961331914595912956131783048380, 9.851949980362829796115950049742, 10.50109878050085543900714568386

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