Properties

Label 2-360-120.77-c1-0-8
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: $\chi_{360} (197, \cdot )$
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

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

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