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Properties

Label 2-360-120.77-c1-0-22
Degree $2$
Conductor $360$
Sign $-0.999 - 0.0394i$
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  + (0.734 − 1.20i)2-s + (−0.922 − 1.77i)4-s + (−2.03 + 0.929i)5-s + (−2.49 − 2.49i)7-s + (−2.82 − 0.187i)8-s + (−0.369 + 3.14i)10-s + 3.92·11-s + (−4.55 − 4.55i)13-s + (−4.84 + 1.18i)14-s + (−2.29 + 3.27i)16-s + (−1.88 + 1.88i)17-s − 4.61·19-s + (3.52 + 2.75i)20-s + (2.88 − 4.74i)22-s + (0.741 + 0.741i)23-s + ⋯
L(s)  = 1  + (0.519 − 0.854i)2-s + (−0.461 − 0.887i)4-s + (−0.909 + 0.415i)5-s + (−0.942 − 0.942i)7-s + (−0.997 − 0.0664i)8-s + (−0.116 + 0.993i)10-s + 1.18·11-s + (−1.26 − 1.26i)13-s + (−1.29 + 0.316i)14-s + (−0.574 + 0.818i)16-s + (−0.457 + 0.457i)17-s − 1.05·19-s + (0.788 + 0.615i)20-s + (0.614 − 1.01i)22-s + (0.154 + 0.154i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0394i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0394i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.999 - 0.0394i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ -0.999 - 0.0394i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0160983 + 0.815523i\)
\(L(\frac12)\) \(\approx\) \(0.0160983 + 0.815523i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.734 + 1.20i)T \)
3 \( 1 \)
5 \( 1 + (2.03 - 0.929i)T \)
good7 \( 1 + (2.49 + 2.49i)T + 7iT^{2} \)
11 \( 1 - 3.92T + 11T^{2} \)
13 \( 1 + (4.55 + 4.55i)T + 13iT^{2} \)
17 \( 1 + (1.88 - 1.88i)T - 17iT^{2} \)
19 \( 1 + 4.61T + 19T^{2} \)
23 \( 1 + (-0.741 - 0.741i)T + 23iT^{2} \)
29 \( 1 + 4.35iT - 29T^{2} \)
31 \( 1 - 9.67T + 31T^{2} \)
37 \( 1 + (-5.39 + 5.39i)T - 37iT^{2} \)
41 \( 1 + 6.33iT - 41T^{2} \)
43 \( 1 + (0.206 + 0.206i)T + 43iT^{2} \)
47 \( 1 + (3.48 - 3.48i)T - 47iT^{2} \)
53 \( 1 + (1.01 - 1.01i)T - 53iT^{2} \)
59 \( 1 + 0.531iT - 59T^{2} \)
61 \( 1 + 3.00iT - 61T^{2} \)
67 \( 1 + (-1.28 + 1.28i)T - 67iT^{2} \)
71 \( 1 - 7.61iT - 71T^{2} \)
73 \( 1 + (0.509 - 0.509i)T - 73iT^{2} \)
79 \( 1 + 1.31iT - 79T^{2} \)
83 \( 1 + (-9.85 + 9.85i)T - 83iT^{2} \)
89 \( 1 + 2.91T + 89T^{2} \)
97 \( 1 + (8.11 + 8.11i)T + 97iT^{2} \)
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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

−10.95978732982263128334031678012, −10.27572492584765039825535762193, −9.513481111047125046142667183944, −8.197782773252228411474724798966, −6.98388958321579348794279153340, −6.16208541359172838937521113953, −4.49536564354391483621209098241, −3.80061347105943180029363082703, −2.70456779288623675714623114982, −0.46093258178860181043786809429, 2.79765034431476052686989873988, 4.16653496213394675413771457961, 4.88173957336280771604114110506, 6.46682912705618659895579514159, 6.80185397022618201667139132430, 8.165269955892494986887740842873, 9.031458428756607544198746047824, 9.580405652682740061538120095896, 11.53996018936501305444362698113, 12.05278357504363740608882251026

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