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Properties

Label 2-585-13.12-c1-0-4
Degree $2$
Conductor $585$
Sign $1$
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  − 2i·2-s − 2·4-s + i·5-s + 3.60i·7-s + 2·10-s + 3i·11-s + 3.60i·13-s + 7.21·14-s − 4·16-s − 3.60·17-s + 7.21i·19-s − 2i·20-s + 6·22-s + 3.60·23-s − 25-s + 7.21·26-s + ⋯
L(s)  = 1  − 1.41i·2-s − 4-s + 0.447i·5-s + 1.36i·7-s + 0.632·10-s + 0.904i·11-s + 0.999i·13-s + 1.92·14-s − 16-s − 0.874·17-s + 1.65i·19-s − 0.447i·20-s + 1.27·22-s + 0.751·23-s − 0.200·25-s + 1.41·26-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $1$
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: \((2,\ 585,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23602\)
\(L(\frac12)\) \(\approx\) \(1.23602\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.60iT \)
good2 \( 1 + 2iT - 2T^{2} \)
7 \( 1 - 3.60iT - 7T^{2} \)
11 \( 1 - 3iT - 11T^{2} \)
17 \( 1 + 3.60T + 17T^{2} \)
19 \( 1 - 7.21iT - 19T^{2} \)
23 \( 1 - 3.60T + 23T^{2} \)
29 \( 1 - 7.21T + 29T^{2} \)
31 \( 1 + 7.21iT - 31T^{2} \)
37 \( 1 - 3.60iT - 37T^{2} \)
41 \( 1 + 11iT - 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + 4iT - 47T^{2} \)
53 \( 1 + 10.8T + 53T^{2} \)
59 \( 1 - 12iT - 59T^{2} \)
61 \( 1 - 13T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 5iT - 71T^{2} \)
73 \( 1 + 7.21iT - 73T^{2} \)
79 \( 1 - 13T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 + 3iT - 89T^{2} \)
97 \( 1 - 3.60iT - 97T^{2} \)
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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

−10.76244194943917901207514640954, −9.951921666342017309018157034326, −9.264104021263137205625715809473, −8.440582805946293332288588797354, −7.03708597777763116281173647207, −6.13476236443244883773583525109, −4.79420707122433485941673250125, −3.76840870973028695816046339357, −2.50548538284112622547594275756, −1.85551754050527559636569970945, 0.70107886146530611510706438638, 3.06659558239422310515133068402, 4.56971035817774309661104182808, 5.15547989339809802879574550380, 6.48528091341689320842539822025, 6.94391595049713855404006931951, 8.021412537110836645874120258115, 8.551689569591561994460434535049, 9.557616754549003930218114164368, 10.83011117654957591658900061588

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