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Properties

Label 2-360-120.53-c1-0-22
Degree $2$
Conductor $360$
Sign $0.220 + 0.975i$
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  + (1.33 − 0.473i)2-s + (1.55 − 1.26i)4-s + (−1.45 − 1.69i)5-s + (1.53 − 1.53i)7-s + (1.47 − 2.41i)8-s + (−2.74 − 1.57i)10-s − 2.72·11-s + (−0.857 + 0.857i)13-s + (1.31 − 2.76i)14-s + (0.818 − 3.91i)16-s + (2.55 + 2.55i)17-s + 3.54·19-s + (−4.39 − 0.803i)20-s + (−3.63 + 1.28i)22-s + (−0.626 + 0.626i)23-s + ⋯
L(s)  = 1  + (0.942 − 0.334i)2-s + (0.776 − 0.630i)4-s + (−0.650 − 0.759i)5-s + (0.578 − 0.578i)7-s + (0.520 − 0.853i)8-s + (−0.866 − 0.498i)10-s − 0.821·11-s + (−0.237 + 0.237i)13-s + (0.351 − 0.738i)14-s + (0.204 − 0.978i)16-s + (0.619 + 0.619i)17-s + 0.812·19-s + (−0.983 − 0.179i)20-s + (−0.774 + 0.274i)22-s + (−0.130 + 0.130i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.220 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.220 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.220 + 0.975i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.220 + 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69015 - 1.35056i\)
\(L(\frac12)\) \(\approx\) \(1.69015 - 1.35056i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.33 + 0.473i)T \)
3 \( 1 \)
5 \( 1 + (1.45 + 1.69i)T \)
good7 \( 1 + (-1.53 + 1.53i)T - 7iT^{2} \)
11 \( 1 + 2.72T + 11T^{2} \)
13 \( 1 + (0.857 - 0.857i)T - 13iT^{2} \)
17 \( 1 + (-2.55 - 2.55i)T + 17iT^{2} \)
19 \( 1 - 3.54T + 19T^{2} \)
23 \( 1 + (0.626 - 0.626i)T - 23iT^{2} \)
29 \( 1 + 5.12iT - 29T^{2} \)
31 \( 1 - 7.89T + 31T^{2} \)
37 \( 1 + (-4.21 - 4.21i)T + 37iT^{2} \)
41 \( 1 - 12.4iT - 41T^{2} \)
43 \( 1 + (5.67 - 5.67i)T - 43iT^{2} \)
47 \( 1 + (9.45 + 9.45i)T + 47iT^{2} \)
53 \( 1 + (-6.46 - 6.46i)T + 53iT^{2} \)
59 \( 1 - 2.51iT - 59T^{2} \)
61 \( 1 - 9.49iT - 61T^{2} \)
67 \( 1 + (9.91 + 9.91i)T + 67iT^{2} \)
71 \( 1 - 2.19iT - 71T^{2} \)
73 \( 1 + (5.71 + 5.71i)T + 73iT^{2} \)
79 \( 1 + 12.7iT - 79T^{2} \)
83 \( 1 + (-3.58 - 3.58i)T + 83iT^{2} \)
89 \( 1 - 10.2T + 89T^{2} \)
97 \( 1 + (1.29 - 1.29i)T - 97iT^{2} \)
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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

−11.67943776211302712260924574375, −10.48309108542729288343873508624, −9.711401994364331970671458678969, −8.109909182027990948742258559773, −7.58878875913903887773805914778, −6.18020953022194006287758477580, −4.97506450710618713276985219840, −4.36523048994677469062231192784, −3.07165599833003541363459053894, −1.28392326376367360080991761305, 2.48130970836442158315265856456, 3.43810550037220053799946581816, 4.84099197842392325085850169831, 5.62042840219375674439838608257, 6.91586844031587533113790424905, 7.68622479038374122712188548059, 8.443799435696119000925469686023, 10.06971103477239995351481469515, 11.01667079090759332609803516964, 11.80122543221977983531328725865

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