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Properties

Label 2-360-8.5-c1-0-2
Degree $2$
Conductor $360$
Sign $-0.844 - 0.535i$
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  + (−0.264 + 1.38i)2-s + (−1.85 − 0.735i)4-s + i·5-s + 0.941·7-s + (1.51 − 2.38i)8-s + (−1.38 − 0.264i)10-s + 4.49i·11-s + 5.55i·13-s + (−0.249 + 1.30i)14-s + (2.91 + 2.73i)16-s − 7.55·17-s − 1.05i·19-s + (0.735 − 1.85i)20-s + (−6.24 − 1.19i)22-s + 1.05·23-s + ⋯
L(s)  = 1  + (−0.187 + 0.982i)2-s + (−0.929 − 0.367i)4-s + 0.447i·5-s + 0.355·7-s + (0.535 − 0.844i)8-s + (−0.439 − 0.0836i)10-s + 1.35i·11-s + 1.54i·13-s + (−0.0665 + 0.349i)14-s + (0.729 + 0.683i)16-s − 1.83·17-s − 0.242i·19-s + (0.164 − 0.415i)20-s + (−1.33 − 0.253i)22-s + 0.220·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.844 - 0.535i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ -0.844 - 0.535i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.268668 + 0.926027i\)
\(L(\frac12)\) \(\approx\) \(0.268668 + 0.926027i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.264 - 1.38i)T \)
3 \( 1 \)
5 \( 1 - iT \)
good7 \( 1 - 0.941T + 7T^{2} \)
11 \( 1 - 4.49iT - 11T^{2} \)
13 \( 1 - 5.55iT - 13T^{2} \)
17 \( 1 + 7.55T + 17T^{2} \)
19 \( 1 + 1.05iT - 19T^{2} \)
23 \( 1 - 1.05T + 23T^{2} \)
29 \( 1 + 2iT - 29T^{2} \)
31 \( 1 - 3.55T + 31T^{2} \)
37 \( 1 - 7.43iT - 37T^{2} \)
41 \( 1 - 3.88T + 41T^{2} \)
43 \( 1 - 1.88iT - 43T^{2} \)
47 \( 1 - 10.0T + 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 + 8.49iT - 59T^{2} \)
61 \( 1 + 8.99iT - 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 - 12.9T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 - 11.5T + 79T^{2} \)
83 \( 1 - 5.88iT - 83T^{2} \)
89 \( 1 - 4.11T + 89T^{2} \)
97 \( 1 - 17.1T + 97T^{2} \)
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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

−11.74585251845096339501436270881, −10.81046730738413672857829590058, −9.670173946243312062247169975660, −9.056683498193185286510730601383, −7.946740285394569677160359550403, −6.86514195281284829144813591041, −6.49741975472866412090327493485, −4.82175164523713210286754866229, −4.24194448948720143536838952211, −2.05213036832515664100053995864, 0.72295684515443759611612403378, 2.49642461781426211753342247840, 3.73585456244142742135525560450, 4.93638819227193315915037266741, 5.94528718389750326083641950583, 7.67116803498985681572197140276, 8.566508516310242305581971136619, 9.103227568623592114200959388004, 10.49185888310233164884772086826, 10.93453907249546685739834675056

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