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Properties

Label 2-360-40.29-c1-0-19
Degree $2$
Conductor $360$
Sign $0.742 + 0.669i$
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 + 0.806i)2-s + (0.699 − 1.87i)4-s + (1.86 − 1.23i)5-s − 0.746i·7-s + (0.699 + 2.74i)8-s + (−1.16 + 2.94i)10-s − 5.36i·11-s − 2.92·13-s + (0.601 + 0.866i)14-s + (−3.02 − 2.62i)16-s + 2.13i·17-s − 1.73i·19-s + (−1.02 − 4.35i)20-s + (4.32 + 6.22i)22-s − 7.49i·23-s + ⋯
L(s)  = 1  + (−0.821 + 0.570i)2-s + (0.349 − 0.936i)4-s + (0.832 − 0.554i)5-s − 0.282i·7-s + (0.247 + 0.968i)8-s + (−0.367 + 0.930i)10-s − 1.61i·11-s − 0.811·13-s + (0.160 + 0.231i)14-s + (−0.755 − 0.655i)16-s + 0.517i·17-s − 0.397i·19-s + (−0.228 − 0.973i)20-s + (0.921 + 1.32i)22-s − 1.56i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.742 + 0.669i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.742 + 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.742 + 0.669i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.742 + 0.669i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.900030 - 0.345606i\)
\(L(\frac12)\) \(\approx\) \(0.900030 - 0.345606i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.16 - 0.806i)T \)
3 \( 1 \)
5 \( 1 + (-1.86 + 1.23i)T \)
good7 \( 1 + 0.746iT - 7T^{2} \)
11 \( 1 + 5.36iT - 11T^{2} \)
13 \( 1 + 2.92T + 13T^{2} \)
17 \( 1 - 2.13iT - 17T^{2} \)
19 \( 1 + 1.73iT - 19T^{2} \)
23 \( 1 + 7.49iT - 23T^{2} \)
29 \( 1 - 6.74iT - 29T^{2} \)
31 \( 1 - 2.64T + 31T^{2} \)
37 \( 1 + 1.07T + 37T^{2} \)
41 \( 1 - 11.2T + 41T^{2} \)
43 \( 1 - 7.44T + 43T^{2} \)
47 \( 1 - 1.73iT - 47T^{2} \)
53 \( 1 + 7.72T + 53T^{2} \)
59 \( 1 - 6.85iT - 59T^{2} \)
61 \( 1 + 6.45iT - 61T^{2} \)
67 \( 1 + 7.44T + 67T^{2} \)
71 \( 1 + 13.2T + 71T^{2} \)
73 \( 1 + 0.690iT - 73T^{2} \)
79 \( 1 + 2.64T + 79T^{2} \)
83 \( 1 - 5.85T + 83T^{2} \)
89 \( 1 - 7.59T + 89T^{2} \)
97 \( 1 - 14.1iT - 97T^{2} \)
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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

−10.85493817311904581555184130229, −10.43545708379377795824026319199, −9.205964937848445264448546077660, −8.723157593482285669661187177150, −7.70724733171965220791502880607, −6.47040176349282608148420638088, −5.76455629105744345379488937079, −4.65905303233235395056769399932, −2.59374407491759750383671509794, −0.889785387306774308798492493614, 1.85786356006178278370954896093, 2.78242270731745568453443177115, 4.40152021646059700186722445524, 5.86090456268283388081922523685, 7.16868571349052082495779464804, 7.65530324250462619357645478437, 9.259004844497631075206854749997, 9.680624454369128778018860597476, 10.37586676460016952010722938899, 11.49204007748723328717376578849

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