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Properties

Label 2-360-40.27-c1-0-19
Degree $2$
Conductor $360$
Sign $-0.135 + 0.990i$
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  + (−1.08 + 0.909i)2-s + (0.345 − 1.96i)4-s + (−0.780 − 2.09i)5-s + (−2.10 + 2.10i)7-s + (1.41 + 2.44i)8-s + (2.75 + 1.55i)10-s + 3.11·11-s + (−2.19 − 2.19i)13-s + (0.366 − 4.20i)14-s + (−3.76 − 1.36i)16-s + (−5.48 − 5.48i)17-s − 3.91i·19-s + (−4.39 + 0.813i)20-s + (−3.37 + 2.83i)22-s + (−2.42 − 2.42i)23-s + ⋯
L(s)  = 1  + (−0.765 + 0.643i)2-s + (0.172 − 0.984i)4-s + (−0.349 − 0.937i)5-s + (−0.796 + 0.796i)7-s + (0.500 + 0.865i)8-s + (0.869 + 0.493i)10-s + 0.940·11-s + (−0.608 − 0.608i)13-s + (0.0978 − 1.12i)14-s + (−0.940 − 0.340i)16-s + (−1.33 − 1.33i)17-s − 0.899i·19-s + (−0.983 + 0.181i)20-s + (−0.720 + 0.604i)22-s + (−0.505 − 0.505i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.135 + 0.990i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ -0.135 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.296842 - 0.340203i\)
\(L(\frac12)\) \(\approx\) \(0.296842 - 0.340203i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.08 - 0.909i)T \)
3 \( 1 \)
5 \( 1 + (0.780 + 2.09i)T \)
good7 \( 1 + (2.10 - 2.10i)T - 7iT^{2} \)
11 \( 1 - 3.11T + 11T^{2} \)
13 \( 1 + (2.19 + 2.19i)T + 13iT^{2} \)
17 \( 1 + (5.48 + 5.48i)T + 17iT^{2} \)
19 \( 1 + 3.91iT - 19T^{2} \)
23 \( 1 + (2.42 + 2.42i)T + 23iT^{2} \)
29 \( 1 + 2.23T + 29T^{2} \)
31 \( 1 + 7.17iT - 31T^{2} \)
37 \( 1 + (-1.23 + 1.23i)T - 37iT^{2} \)
41 \( 1 + 4.40T + 41T^{2} \)
43 \( 1 + (1.50 - 1.50i)T - 43iT^{2} \)
47 \( 1 + (-2.00 + 2.00i)T - 47iT^{2} \)
53 \( 1 + (-5.56 - 5.56i)T + 53iT^{2} \)
59 \( 1 - 5.44iT - 59T^{2} \)
61 \( 1 + 7.46iT - 61T^{2} \)
67 \( 1 + (-6.40 - 6.40i)T + 67iT^{2} \)
71 \( 1 - 2.00iT - 71T^{2} \)
73 \( 1 + (1.49 - 1.49i)T - 73iT^{2} \)
79 \( 1 - 1.91T + 79T^{2} \)
83 \( 1 + (-6.67 + 6.67i)T - 83iT^{2} \)
89 \( 1 - 10.0iT - 89T^{2} \)
97 \( 1 + (10.0 + 10.0i)T + 97iT^{2} \)
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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

−11.25206692105713354847646602249, −9.793938990298187340434411271622, −9.206979054250768515948526484022, −8.628409761373963520053093882259, −7.43766956491780047619414149670, −6.51311061630151311899109823453, −5.46295731524143212612514906181, −4.42857044187458427487236606607, −2.43029026753941095511293994149, −0.37965394699664438392654909508, 1.91913906810696340643963994584, 3.51321186942508232967767308597, 4.08849152237613092542789348326, 6.49419425928900522976346106161, 6.91441875357068584637621892460, 8.019802998924720848628738055157, 9.100049073106131558043633179254, 10.05715474102669231480987901670, 10.60965961135222999203158295795, 11.54376028266928904522869166287

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