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Properties

Label 2-360-120.53-c1-0-4
Degree $2$
Conductor $360$
Sign $0.295 - 0.955i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.295 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.295 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.295 - 0.955i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.295 - 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.551564 + 0.406578i\)
\(L(\frac12)\) \(\approx\) \(0.551564 + 0.406578i\)
\(L(\frac{3}{2})\) 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

−11.65483775959075629476900778037, −10.40961418270601594402564228854, −9.889452803992486932111625786462, −9.198266347193275096754810894950, −8.149595795706716740526203743905, −6.90216688993879829472090634528, −5.85533030643751219614852655882, −4.51727163731297025588936448189, −2.79594379792712907285878752470, −2.23171311787756952024245442011, 0.52275934961900694535809615308, 2.67346273457295490290166311958, 4.64898728654451289499232986600, 5.52115342618482815548215475129, 6.51620826954414626186352745309, 7.49921036541805853177694624507, 8.316540717300303949883522470217, 9.594306761035104831711219522600, 10.14036832630279911344140738220, 10.61615246766468679664654896783

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