Properties

Label 2-360-72.59-c1-0-21
Degree $2$
Conductor $360$
Sign $0.974 - 0.223i$
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  + (0.0218 + 1.41i)2-s + (−0.452 − 1.67i)3-s + (−1.99 + 0.0617i)4-s + (−0.5 + 0.866i)5-s + (2.35 − 0.675i)6-s + (0.550 − 0.317i)7-s + (−0.131 − 2.82i)8-s + (−2.59 + 1.51i)9-s + (−1.23 − 0.688i)10-s + (4.49 − 2.59i)11-s + (1.00 + 3.31i)12-s + (5.73 + 3.31i)13-s + (0.461 + 0.771i)14-s + (1.67 + 0.444i)15-s + (3.99 − 0.247i)16-s − 4.95i·17-s + ⋯
L(s)  = 1  + (0.0154 + 0.999i)2-s + (−0.261 − 0.965i)3-s + (−0.999 + 0.0308i)4-s + (−0.223 + 0.387i)5-s + (0.961 − 0.275i)6-s + (0.208 − 0.120i)7-s + (−0.0463 − 0.998i)8-s + (−0.863 + 0.504i)9-s + (−0.390 − 0.217i)10-s + (1.35 − 0.782i)11-s + (0.290 + 0.956i)12-s + (1.59 + 0.918i)13-s + (0.123 + 0.206i)14-s + (0.432 + 0.114i)15-s + (0.998 − 0.0617i)16-s − 1.20i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.223i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 - 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.974 - 0.223i$
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: \((2,\ 360,\ (\ :1/2),\ 0.974 - 0.223i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17588 + 0.133152i\)
\(L(\frac12)\) \(\approx\) \(1.17588 + 0.133152i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0218 - 1.41i)T \)
3 \( 1 + (0.452 + 1.67i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
good7 \( 1 + (-0.550 + 0.317i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.49 + 2.59i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-5.73 - 3.31i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.95iT - 17T^{2} \)
19 \( 1 + 0.264T + 19T^{2} \)
23 \( 1 + (-3.14 + 5.45i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.68 - 2.91i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.81 - 1.04i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 2.07iT - 37T^{2} \)
41 \( 1 + (9.24 + 5.33i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.19 - 3.80i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.26 - 5.65i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 3.24T + 53T^{2} \)
59 \( 1 + (0.776 + 0.448i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.09 - 3.51i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.19 + 9.00i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 16.0T + 71T^{2} \)
73 \( 1 - 8.48T + 73T^{2} \)
79 \( 1 + (-2.27 + 1.31i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.82 + 5.67i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 9.84iT - 89T^{2} \)
97 \( 1 + (-2.80 - 4.86i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.53220713005554362786973032193, −10.79107368173058324668085623571, −9.051781230519788013812005239365, −8.649460858099560367128619926015, −7.47952597161253777870801144399, −6.54370251178593934872010699572, −6.20222991839152102448122519693, −4.70738763629981138773247037864, −3.38471753491721647602636788875, −1.09603728569796977619380615065, 1.37961390122765958684445990645, 3.45567002299755175035620073062, 4.08143662345896593187288583940, 5.20497859879895335700975353162, 6.25305072293065508017883297064, 8.218445502819398180187183097544, 8.838433504776365379860888505696, 9.745335684085962530602819283555, 10.55105246444258147528972603261, 11.40946404645013298942652459489

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