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Properties

Label 2-360-40.27-c1-0-11
Degree $2$
Conductor $360$
Sign $0.879 + 0.476i$
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  + (−1.31 − 0.518i)2-s + (1.46 + 1.36i)4-s + (2.22 − 0.169i)5-s + (0.645 − 0.645i)7-s + (−1.21 − 2.55i)8-s + (−3.02 − 0.932i)10-s + 2.11·11-s + (−1.65 − 1.65i)13-s + (−1.18 + 0.514i)14-s + (0.279 + 3.99i)16-s + (4.23 + 4.23i)17-s + 2.18i·19-s + (3.49 + 2.79i)20-s + (−2.78 − 1.09i)22-s + (−6.05 − 6.05i)23-s + ⋯
L(s)  = 1  + (−0.930 − 0.366i)2-s + (0.731 + 0.681i)4-s + (0.997 − 0.0757i)5-s + (0.243 − 0.243i)7-s + (−0.430 − 0.902i)8-s + (−0.955 − 0.294i)10-s + 0.639·11-s + (−0.458 − 0.458i)13-s + (−0.316 + 0.137i)14-s + (0.0697 + 0.997i)16-s + (1.02 + 1.02i)17-s + 0.502i·19-s + (0.780 + 0.624i)20-s + (−0.594 − 0.234i)22-s + (−1.26 − 1.26i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 + 0.476i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.879 + 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.879 + 0.476i$
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: \((2,\ 360,\ (\ :1/2),\ 0.879 + 0.476i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07734 - 0.272993i\)
\(L(\frac12)\) \(\approx\) \(1.07734 - 0.272993i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.31 + 0.518i)T \)
3 \( 1 \)
5 \( 1 + (-2.22 + 0.169i)T \)
good7 \( 1 + (-0.645 + 0.645i)T - 7iT^{2} \)
11 \( 1 - 2.11T + 11T^{2} \)
13 \( 1 + (1.65 + 1.65i)T + 13iT^{2} \)
17 \( 1 + (-4.23 - 4.23i)T + 17iT^{2} \)
19 \( 1 - 2.18iT - 19T^{2} \)
23 \( 1 + (6.05 + 6.05i)T + 23iT^{2} \)
29 \( 1 - 7.93T + 29T^{2} \)
31 \( 1 - 0.574iT - 31T^{2} \)
37 \( 1 + (-6.90 + 6.90i)T - 37iT^{2} \)
41 \( 1 + 2.99T + 41T^{2} \)
43 \( 1 + (-3.18 + 3.18i)T - 43iT^{2} \)
47 \( 1 + (5.04 - 5.04i)T - 47iT^{2} \)
53 \( 1 + (1.19 + 1.19i)T + 53iT^{2} \)
59 \( 1 + 11.5iT - 59T^{2} \)
61 \( 1 - 2.35iT - 61T^{2} \)
67 \( 1 + (-4.99 - 4.99i)T + 67iT^{2} \)
71 \( 1 - 5.01iT - 71T^{2} \)
73 \( 1 + (6.18 - 6.18i)T - 73iT^{2} \)
79 \( 1 + 10.1T + 79T^{2} \)
83 \( 1 + (11.7 - 11.7i)T - 83iT^{2} \)
89 \( 1 - 8.65iT - 89T^{2} \)
97 \( 1 + (8.06 + 8.06i)T + 97iT^{2} \)
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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

−11.10862831821577877845724922646, −10.14569685346951394896231510093, −9.841796678362584916504889317740, −8.595316704099452988468635753308, −7.908260513220583920994170345307, −6.62781876540904512204903509470, −5.80162688799296410639631342825, −4.12455156001329025779625763273, −2.61677392127558347877708746241, −1.30679271246968143933603438092, 1.44906092712275154863643910101, 2.78051359269115281999708574816, 4.91127506560655807643280223885, 5.91929748759307969705747910330, 6.79451111865127351362055501570, 7.79683647999140253972089360464, 8.889936462199550214227401939475, 9.713246642903625450962982669917, 10.14319861523791171825908113244, 11.54687747858550199129113006019

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