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Properties

Label 2-360-120.53-c1-0-16
Degree $2$
Conductor $360$
Sign $0.714 - 0.699i$
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  + (1.16 + 0.803i)2-s + (0.709 + 1.87i)4-s + (1.29 − 1.82i)5-s + (−0.306 + 0.306i)7-s + (−0.676 + 2.74i)8-s + (2.97 − 1.08i)10-s + 4.06·11-s + (0.625 − 0.625i)13-s + (−0.603 + 0.110i)14-s + (−2.99 + 2.65i)16-s + (3.57 + 3.57i)17-s − 6.82·19-s + (4.32 + 1.12i)20-s + (4.73 + 3.26i)22-s + (−1.58 + 1.58i)23-s + ⋯
L(s)  = 1  + (0.822 + 0.568i)2-s + (0.354 + 0.935i)4-s + (0.578 − 0.815i)5-s + (−0.115 + 0.115i)7-s + (−0.239 + 0.970i)8-s + (0.939 − 0.343i)10-s + 1.22·11-s + (0.173 − 0.173i)13-s + (−0.161 + 0.0295i)14-s + (−0.748 + 0.663i)16-s + (0.867 + 0.867i)17-s − 1.56·19-s + (0.967 + 0.251i)20-s + (1.00 + 0.696i)22-s + (−0.331 + 0.331i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 - 0.699i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.714 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.714 - 0.699i$
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: \((2,\ 360,\ (\ :1/2),\ 0.714 - 0.699i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.12954 + 0.868779i\)
\(L(\frac12)\) \(\approx\) \(2.12954 + 0.868779i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.16 - 0.803i)T \)
3 \( 1 \)
5 \( 1 + (-1.29 + 1.82i)T \)
good7 \( 1 + (0.306 - 0.306i)T - 7iT^{2} \)
11 \( 1 - 4.06T + 11T^{2} \)
13 \( 1 + (-0.625 + 0.625i)T - 13iT^{2} \)
17 \( 1 + (-3.57 - 3.57i)T + 17iT^{2} \)
19 \( 1 + 6.82T + 19T^{2} \)
23 \( 1 + (1.58 - 1.58i)T - 23iT^{2} \)
29 \( 1 + 8.50iT - 29T^{2} \)
31 \( 1 + 2.56T + 31T^{2} \)
37 \( 1 + (1.69 + 1.69i)T + 37iT^{2} \)
41 \( 1 + 5.19iT - 41T^{2} \)
43 \( 1 + (4.18 - 4.18i)T - 43iT^{2} \)
47 \( 1 + (-4.58 - 4.58i)T + 47iT^{2} \)
53 \( 1 + (7.41 + 7.41i)T + 53iT^{2} \)
59 \( 1 - 8.79iT - 59T^{2} \)
61 \( 1 + 6.08iT - 61T^{2} \)
67 \( 1 + (6.18 + 6.18i)T + 67iT^{2} \)
71 \( 1 + 14.7iT - 71T^{2} \)
73 \( 1 + (-3.05 - 3.05i)T + 73iT^{2} \)
79 \( 1 + 8.56iT - 79T^{2} \)
83 \( 1 + (-5.13 - 5.13i)T + 83iT^{2} \)
89 \( 1 + 9.88T + 89T^{2} \)
97 \( 1 + (-1.75 + 1.75i)T - 97iT^{2} \)
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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

−11.96956273574504291504166356948, −10.77026746729583205865956039527, −9.515466176667216218470500872223, −8.637607135352974085695573493842, −7.79386839102462578610637264810, −6.31493035079758313329779104534, −5.93173424048146827938613971620, −4.59495089096359839840430377575, −3.70523777322112034633367453714, −1.92067973328556760878445612208, 1.68933124662226169449339701436, 3.05718981795471473876333546104, 4.08754248319983243902105627159, 5.42070124972916454279330331641, 6.47909080380202478823359351746, 7.03294920235932742771914517906, 8.829070920154491527971046234498, 9.795070786956172998198976267243, 10.54721740870818297897317477754, 11.36325902809832438552868187363

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