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Properties

Label 2-360-120.77-c1-0-14
Degree $2$
Conductor $360$
Sign $-0.743 + 0.668i$
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  + (−1.40 − 0.159i)2-s + (1.94 + 0.449i)4-s + (−0.215 + 2.22i)5-s + (−2.32 − 2.32i)7-s + (−2.66 − 0.943i)8-s + (0.658 − 3.09i)10-s − 5.57·11-s + (−1.79 − 1.79i)13-s + (2.88 + 3.63i)14-s + (3.59 + 1.75i)16-s + (5.56 − 5.56i)17-s + 2.61·19-s + (−1.42 + 4.24i)20-s + (7.83 + 0.892i)22-s + (−4.44 − 4.44i)23-s + ⋯
L(s)  = 1  + (−0.993 − 0.113i)2-s + (0.974 + 0.224i)4-s + (−0.0963 + 0.995i)5-s + (−0.877 − 0.877i)7-s + (−0.942 − 0.333i)8-s + (0.208 − 0.978i)10-s − 1.68·11-s + (−0.497 − 0.497i)13-s + (0.772 + 0.970i)14-s + (0.898 + 0.438i)16-s + (1.34 − 1.34i)17-s + 0.600·19-s + (−0.317 + 0.948i)20-s + (1.67 + 0.190i)22-s + (−0.927 − 0.927i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.743 + 0.668i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.743 + 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.743 + 0.668i$
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: \((2,\ 360,\ (\ :1/2),\ -0.743 + 0.668i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0914176 - 0.238342i\)
\(L(\frac12)\) \(\approx\) \(0.0914176 - 0.238342i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.159i)T \)
3 \( 1 \)
5 \( 1 + (0.215 - 2.22i)T \)
good7 \( 1 + (2.32 + 2.32i)T + 7iT^{2} \)
11 \( 1 + 5.57T + 11T^{2} \)
13 \( 1 + (1.79 + 1.79i)T + 13iT^{2} \)
17 \( 1 + (-5.56 + 5.56i)T - 17iT^{2} \)
19 \( 1 - 2.61T + 19T^{2} \)
23 \( 1 + (4.44 + 4.44i)T + 23iT^{2} \)
29 \( 1 + 2.12iT - 29T^{2} \)
31 \( 1 + 4.52T + 31T^{2} \)
37 \( 1 + (5.02 - 5.02i)T - 37iT^{2} \)
41 \( 1 - 1.73iT - 41T^{2} \)
43 \( 1 + (1.19 + 1.19i)T + 43iT^{2} \)
47 \( 1 + (0.849 - 0.849i)T - 47iT^{2} \)
53 \( 1 + (4.22 - 4.22i)T - 53iT^{2} \)
59 \( 1 + 8.08iT - 59T^{2} \)
61 \( 1 + 3.13iT - 61T^{2} \)
67 \( 1 + (-1.86 + 1.86i)T - 67iT^{2} \)
71 \( 1 - 6.95iT - 71T^{2} \)
73 \( 1 + (5.86 - 5.86i)T - 73iT^{2} \)
79 \( 1 - 2.52iT - 79T^{2} \)
83 \( 1 + (0.694 - 0.694i)T - 83iT^{2} \)
89 \( 1 - 12.1T + 89T^{2} \)
97 \( 1 + (-4.09 - 4.09i)T + 97iT^{2} \)
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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

−10.69761728121360086502543901050, −10.11507091671326186944477575655, −9.694220787557290595973596612956, −7.960770147394724536409543092518, −7.51786248322671674395686591357, −6.66207397273406655529293096723, −5.41166618733498662652453095405, −3.36665086954477138677987624859, −2.64860049680412012441124654006, −0.22024175967457437793251276673, 1.89597934739094380057688432968, 3.34504066673928518213776373018, 5.37704968827377651153168446617, 5.86594409203414438868085584762, 7.44880099068407130819713696105, 8.117555703606824199168785763481, 9.074072370201616024496319887503, 9.806169029029175497009469071790, 10.55417383044400353932227919620, 11.94205676039142037351649840687

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