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Properties

Label 2-360-40.3-c1-0-17
Degree $2$
Conductor $360$
Sign $0.865 + 0.501i$
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  + (−1.17 + 0.788i)2-s + (0.758 − 1.85i)4-s + (0.386 − 2.20i)5-s + (1.51 + 1.51i)7-s + (0.568 + 2.77i)8-s + (1.28 + 2.89i)10-s − 3.92·11-s + (3.56 − 3.56i)13-s + (−2.97 − 0.585i)14-s + (−2.85 − 2.80i)16-s + (1.37 − 1.37i)17-s − 4i·19-s + (−3.78 − 2.38i)20-s + (4.60 − 3.09i)22-s + (5.17 − 5.17i)23-s + ⋯
L(s)  = 1  + (−0.830 + 0.557i)2-s + (0.379 − 0.925i)4-s + (0.172 − 0.984i)5-s + (0.573 + 0.573i)7-s + (0.200 + 0.979i)8-s + (0.405 + 0.914i)10-s − 1.18·11-s + (0.988 − 0.988i)13-s + (−0.795 − 0.156i)14-s + (−0.712 − 0.701i)16-s + (0.333 − 0.333i)17-s − 0.917i·19-s + (−0.846 − 0.533i)20-s + (0.982 − 0.659i)22-s + (1.07 − 1.07i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.501i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.865 + 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.865 + 0.501i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.865 + 0.501i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.915340 - 0.245902i\)
\(L(\frac12)\) \(\approx\) \(0.915340 - 0.245902i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.17 - 0.788i)T \)
3 \( 1 \)
5 \( 1 + (-0.386 + 2.20i)T \)
good7 \( 1 + (-1.51 - 1.51i)T + 7iT^{2} \)
11 \( 1 + 3.92T + 11T^{2} \)
13 \( 1 + (-3.56 + 3.56i)T - 13iT^{2} \)
17 \( 1 + (-1.37 + 1.37i)T - 17iT^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + (-5.17 + 5.17i)T - 23iT^{2} \)
29 \( 1 - 5.95T + 29T^{2} \)
31 \( 1 - 7.12iT - 31T^{2} \)
37 \( 1 + (3.56 + 3.56i)T + 37iT^{2} \)
41 \( 1 + 2.75T + 41T^{2} \)
43 \( 1 + (-5.40 - 5.40i)T + 43iT^{2} \)
47 \( 1 + (-1.54 - 1.54i)T + 47iT^{2} \)
53 \( 1 + (-1.81 + 1.81i)T - 53iT^{2} \)
59 \( 1 + 3.92iT - 59T^{2} \)
61 \( 1 + 13.1iT - 61T^{2} \)
67 \( 1 + (5.40 - 5.40i)T - 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-5 - 5i)T + 73iT^{2} \)
79 \( 1 + 7.12T + 79T^{2} \)
83 \( 1 + (-6.67 - 6.67i)T + 83iT^{2} \)
89 \( 1 - 18.4iT - 89T^{2} \)
97 \( 1 + (10.4 - 10.4i)T - 97iT^{2} \)
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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

−11.02711503412780394046785675764, −10.43570250100718122117104700712, −9.237778344984600984035197184913, −8.461381002744791956538687676247, −8.003153952068009179488366320681, −6.65231743211343330427762610902, −5.35826696956636771364557704612, −4.99148402998226119120413266306, −2.62990210467620933289652238909, −0.937107379555204188875558241873, 1.62171367303887399548117714078, 3.02004917380948074744113320179, 4.14348012234608001858585007004, 5.89829992397567927094524429153, 7.11561611638571358508002474594, 7.78125227812661400963024391747, 8.762086798998209864489830407878, 9.960230583099411557121026075208, 10.60276943205098565138475147784, 11.21365904686143832076738547181

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