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Properties

Label 2-585-13.12-c1-0-7
Degree $2$
Conductor $585$
Sign $0.899 - 0.435i$
Analytic cond. $4.67124$
Root an. cond. $2.16130$
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  − 0.571i·2-s + 1.67·4-s + i·5-s + 2.67i·7-s − 2.10i·8-s + 0.571·10-s + 5.10i·11-s + (−1.57 − 3.24i)13-s + 1.52·14-s + 2.14·16-s + 5.34·17-s + 6.24i·19-s + 1.67i·20-s + 2.91·22-s − 2.42·23-s + ⋯
L(s)  = 1  − 0.404i·2-s + 0.836·4-s + 0.447i·5-s + 1.01i·7-s − 0.742i·8-s + 0.180·10-s + 1.53i·11-s + (−0.435 − 0.899i)13-s + 0.408·14-s + 0.535·16-s + 1.29·17-s + 1.43i·19-s + 0.374i·20-s + 0.622·22-s − 0.506·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.899 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.899 - 0.435i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ 0.899 - 0.435i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73472 + 0.398076i\)
\(L(\frac12)\) \(\approx\) \(1.73472 + 0.398076i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 - iT \)
13 \( 1 + (1.57 + 3.24i)T \)
good2 \( 1 + 0.571iT - 2T^{2} \)
7 \( 1 - 2.67iT - 7T^{2} \)
11 \( 1 - 5.10iT - 11T^{2} \)
17 \( 1 - 5.34T + 17T^{2} \)
19 \( 1 - 6.24iT - 19T^{2} \)
23 \( 1 + 2.42T + 23T^{2} \)
29 \( 1 + 2.67T + 29T^{2} \)
31 \( 1 + 0.244iT - 31T^{2} \)
37 \( 1 + 3.32iT - 37T^{2} \)
41 \( 1 + 6.48iT - 41T^{2} \)
43 \( 1 - 10.9T + 43T^{2} \)
47 \( 1 - 2.67iT - 47T^{2} \)
53 \( 1 - 4.20T + 53T^{2} \)
59 \( 1 + 0.899iT - 59T^{2} \)
61 \( 1 + 5.81T + 61T^{2} \)
67 \( 1 - 2.18iT - 67T^{2} \)
71 \( 1 + 6.24iT - 71T^{2} \)
73 \( 1 + 10.9iT - 73T^{2} \)
79 \( 1 + 3.63T + 79T^{2} \)
83 \( 1 + 9.81iT - 83T^{2} \)
89 \( 1 - 7.63iT - 89T^{2} \)
97 \( 1 + 11.3iT - 97T^{2} \)
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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

−10.56710850692038133678719090505, −10.13078100564380843607422962160, −9.300563707083350697684080532871, −7.76116441808582329921639483856, −7.43888584070370796884369906750, −6.10034364224783362986152753324, −5.44721569294066927441226191490, −3.84379175514867504841765328702, −2.70949790635949699963224549918, −1.81133743791115713385531021838, 1.07320586777369115378600175713, 2.76202744438686352046787282656, 3.97719157037742959725082459355, 5.25315172997046177771128233011, 6.17282574678201334675619969308, 7.10039861876992353778510819512, 7.81914933086706114838632693375, 8.736599653825606678427711008095, 9.784245511605885976875999837896, 10.80310358016388733284902815987

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