Properties

Label 2-360-72.59-c1-0-19
Degree $2$
Conductor $360$
Sign $-0.436 + 0.899i$
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  + (−1.27 − 0.603i)2-s + (−1.41 + 0.992i)3-s + (1.27 + 1.54i)4-s + (−0.5 + 0.866i)5-s + (2.41 − 0.411i)6-s + (−4.17 + 2.40i)7-s + (−0.694 − 2.74i)8-s + (1.03 − 2.81i)9-s + (1.16 − 0.805i)10-s + (−0.689 + 0.397i)11-s + (−3.33 − 0.930i)12-s + (0.795 + 0.459i)13-s + (6.78 − 0.562i)14-s + (−0.149 − 1.72i)15-s + (−0.767 + 3.92i)16-s − 6.05i·17-s + ⋯
L(s)  = 1  + (−0.904 − 0.426i)2-s + (−0.819 + 0.572i)3-s + (0.635 + 0.771i)4-s + (−0.223 + 0.387i)5-s + (0.985 − 0.168i)6-s + (−1.57 + 0.910i)7-s + (−0.245 − 0.969i)8-s + (0.343 − 0.939i)9-s + (0.367 − 0.254i)10-s + (−0.207 + 0.119i)11-s + (−0.963 − 0.268i)12-s + (0.220 + 0.127i)13-s + (1.81 − 0.150i)14-s + (−0.0385 − 0.445i)15-s + (−0.191 + 0.981i)16-s − 1.46i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.436 + 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.436 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.436 + 0.899i$
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: \((2,\ 360,\ (\ :1/2),\ -0.436 + 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0817587 - 0.130615i\)
\(L(\frac12)\) \(\approx\) \(0.0817587 - 0.130615i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.27 + 0.603i)T \)
3 \( 1 + (1.41 - 0.992i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
good7 \( 1 + (4.17 - 2.40i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.689 - 0.397i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.795 - 0.459i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 6.05iT - 17T^{2} \)
19 \( 1 - 3.42T + 19T^{2} \)
23 \( 1 + (-2.37 + 4.11i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.66 + 2.89i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (7.93 + 4.57i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.14iT - 37T^{2} \)
41 \( 1 + (6.46 + 3.73i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.23 + 5.60i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.567 + 0.983i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 3.88T + 53T^{2} \)
59 \( 1 + (-6.75 - 3.89i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.22 + 1.28i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.832 + 1.44i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.06T + 71T^{2} \)
73 \( 1 - 9.49T + 73T^{2} \)
79 \( 1 + (0.909 - 0.525i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (10.9 - 6.32i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 15.3iT - 89T^{2} \)
97 \( 1 + (2.93 + 5.08i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.14060015156158979534630233947, −10.01683130591079100181665788681, −9.584540987263551800310861220090, −8.760812167761299446993941937266, −7.17938590343967234695760454335, −6.53463337083523072247142054631, −5.39275847639742790380838913998, −3.66538111467024167229641702973, −2.72650141350749706293504030281, −0.16368732378404712408167839039, 1.34805561426606265242048124743, 3.50626501599396558391753594979, 5.30800915956491399877530969866, 6.22061399077647657632866712413, 7.04827686074522413108986939870, 7.75819295795915924679753460694, 8.976594530088731312297135317929, 9.991826505298199783544970307866, 10.65849918265698676253008723985, 11.50179858684405587692334321002

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