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Properties

Label 2-360-72.59-c1-0-36
Degree $2$
Conductor $360$
Sign $-0.166 + 0.986i$
Analytic cond. $2.87461$
Root an. cond. $1.69546$
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  + (0.962 − 1.03i)2-s + (−1.57 + 0.720i)3-s + (−0.146 − 1.99i)4-s + (−0.5 + 0.866i)5-s + (−0.770 + 2.32i)6-s + (1.68 − 0.970i)7-s + (−2.20 − 1.76i)8-s + (1.96 − 2.26i)9-s + (0.415 + 1.35i)10-s + (4.90 − 2.83i)11-s + (1.66 + 3.03i)12-s + (−4.57 − 2.63i)13-s + (0.612 − 2.67i)14-s + (0.164 − 1.72i)15-s + (−3.95 + 0.584i)16-s − 1.68i·17-s + ⋯
L(s)  = 1  + (0.680 − 0.732i)2-s + (−0.909 + 0.415i)3-s + (−0.0733 − 0.997i)4-s + (−0.223 + 0.387i)5-s + (−0.314 + 0.949i)6-s + (0.635 − 0.366i)7-s + (−0.780 − 0.625i)8-s + (0.654 − 0.756i)9-s + (0.131 + 0.427i)10-s + (1.47 − 0.853i)11-s + (0.481 + 0.876i)12-s + (−1.26 − 0.731i)13-s + (0.163 − 0.715i)14-s + (0.0423 − 0.445i)15-s + (−0.989 + 0.146i)16-s − 0.409i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.166 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.166 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.166 + 0.986i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ -0.166 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.892450 - 1.05535i\)
\(L(\frac12)\) \(\approx\) \(0.892450 - 1.05535i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.962 + 1.03i)T \)
3 \( 1 + (1.57 - 0.720i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
good7 \( 1 + (-1.68 + 0.970i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.90 + 2.83i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.57 + 2.63i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.68iT - 17T^{2} \)
19 \( 1 + 1.99T + 19T^{2} \)
23 \( 1 + (-3.68 + 6.38i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.72 + 2.98i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-9.12 - 5.26i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 8.70iT - 37T^{2} \)
41 \( 1 + (0.985 + 0.569i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.18 - 3.78i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.78 + 3.09i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 2.76T + 53T^{2} \)
59 \( 1 + (-4.64 - 2.68i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.02 + 4.05i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.52 - 4.36i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.47T + 71T^{2} \)
73 \( 1 + 14.6T + 73T^{2} \)
79 \( 1 + (-2.53 + 1.46i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.94 - 2.85i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 11.1iT - 89T^{2} \)
97 \( 1 + (-1.57 - 2.72i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.38917236936377313289945921417, −10.47641840759406933847691719621, −9.851387275038663258429968378067, −8.605552409412312920273725829897, −6.95139078246845327890123726590, −6.21158404052415289117516471225, −4.95010608729250652990828521246, −4.27255314988109289070747117642, −2.97284341545090465888921602560, −0.923285329697265112725271582511, 1.92907323496977799489895295235, 4.16143927098908898438234449874, 4.83023612083815145161567327189, 5.85183587366188615690098086418, 6.94400865905555404068316220295, 7.49006767304204312901022495781, 8.745491329692993790167781345673, 9.718558582869988305228355620359, 11.38149767916693050773668037281, 11.84758933035815313702441667222

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