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Properties

Label 2-360-40.27-c1-0-25
Degree $2$
Conductor $360$
Sign $0.729 + 0.684i$
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 − 0.109i)2-s + (1.97 − 0.308i)4-s + (−0.0696 − 2.23i)5-s + (1.21 − 1.21i)7-s + (2.75 − 0.650i)8-s + (−0.342 − 3.14i)10-s − 5.23·11-s + (0.361 + 0.361i)13-s + (1.58 − 1.85i)14-s + (3.80 − 1.21i)16-s + (1.66 + 1.66i)17-s + 3.72i·19-s + (−0.826 − 4.39i)20-s + (−7.38 + 0.572i)22-s + (3.17 + 3.17i)23-s + ⋯
L(s)  = 1  + (0.997 − 0.0773i)2-s + (0.988 − 0.154i)4-s + (−0.0311 − 0.999i)5-s + (0.460 − 0.460i)7-s + (0.973 − 0.230i)8-s + (−0.108 − 0.994i)10-s − 1.57·11-s + (0.100 + 0.100i)13-s + (0.423 − 0.494i)14-s + (0.952 − 0.304i)16-s + (0.403 + 0.403i)17-s + 0.855i·19-s + (−0.184 − 0.982i)20-s + (−1.57 + 0.122i)22-s + (0.662 + 0.662i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 + 0.684i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.729 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.729 + 0.684i$
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: \((2,\ 360,\ (\ :1/2),\ 0.729 + 0.684i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.26127 - 0.895162i\)
\(L(\frac12)\) \(\approx\) \(2.26127 - 0.895162i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 + 0.109i)T \)
3 \( 1 \)
5 \( 1 + (0.0696 + 2.23i)T \)
good7 \( 1 + (-1.21 + 1.21i)T - 7iT^{2} \)
11 \( 1 + 5.23T + 11T^{2} \)
13 \( 1 + (-0.361 - 0.361i)T + 13iT^{2} \)
17 \( 1 + (-1.66 - 1.66i)T + 17iT^{2} \)
19 \( 1 - 3.72iT - 19T^{2} \)
23 \( 1 + (-3.17 - 3.17i)T + 23iT^{2} \)
29 \( 1 - 6.70T + 29T^{2} \)
31 \( 1 + 2.74iT - 31T^{2} \)
37 \( 1 + (4.13 - 4.13i)T - 37iT^{2} \)
41 \( 1 - 7.40T + 41T^{2} \)
43 \( 1 + (5.67 - 5.67i)T - 43iT^{2} \)
47 \( 1 + (7.31 - 7.31i)T - 47iT^{2} \)
53 \( 1 + (10.0 + 10.0i)T + 53iT^{2} \)
59 \( 1 - 11.3iT - 59T^{2} \)
61 \( 1 + 13.3iT - 61T^{2} \)
67 \( 1 + (5.40 + 5.40i)T + 67iT^{2} \)
71 \( 1 + 2.63iT - 71T^{2} \)
73 \( 1 + (-2.67 + 2.67i)T - 73iT^{2} \)
79 \( 1 + 5.91T + 79T^{2} \)
83 \( 1 + (-0.742 + 0.742i)T - 83iT^{2} \)
89 \( 1 + 1.75iT - 89T^{2} \)
97 \( 1 + (-6.64 - 6.64i)T + 97iT^{2} \)
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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

−11.47165218963412200842102245550, −10.60635981471504061180008363152, −9.738297532361091725254433833147, −8.123896040317800116991792178510, −7.72585992307713356720243210046, −6.21390386846128406970362682608, −5.17667485404866495863965287878, −4.53852168250340971475603089337, −3.16857234167508674404126260265, −1.52640731907902347530186811905, 2.40137039991865164312157839337, 3.15045727823877698805411309492, 4.75983858275643936976816843736, 5.54193726087855968034699937539, 6.71986246112697845218401324926, 7.52246670857984481702168878371, 8.497777324269708121693337747367, 10.17823189231291271590649512554, 10.78331879249942319395244976763, 11.56942898123444605113740250360

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