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Properties

Label 2-585-13.4-c1-0-11
Degree $2$
Conductor $585$
Sign $-0.252 - 0.967i$
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  + (2.36 + 1.36i)2-s + (2.73 + 4.73i)4-s − i·5-s + (−2.13 + 1.23i)7-s + 9.46i·8-s + (1.36 − 2.36i)10-s + (3 + 1.73i)11-s + (−0.866 − 3.5i)13-s − 6.73·14-s + (−7.46 + 12.9i)16-s + (1.63 + 2.83i)17-s + (−1.26 + 0.732i)19-s + (4.73 − 2.73i)20-s + (4.73 + 8.19i)22-s + (3.73 − 6.46i)23-s + ⋯
L(s)  = 1  + (1.67 + 0.965i)2-s + (1.36 + 2.36i)4-s − 0.447i·5-s + (−0.806 + 0.465i)7-s + 3.34i·8-s + (0.431 − 0.748i)10-s + (0.904 + 0.522i)11-s + (−0.240 − 0.970i)13-s − 1.79·14-s + (−1.86 + 3.23i)16-s + (0.396 + 0.686i)17-s + (−0.290 + 0.167i)19-s + (1.05 − 0.610i)20-s + (1.00 + 1.74i)22-s + (0.778 − 1.34i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.252 - 0.967i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -0.252 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.16323 + 2.80054i\)
\(L(\frac12)\) \(\approx\) \(2.16323 + 2.80054i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 3.5i)T \)
good2 \( 1 + (-2.36 - 1.36i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (2.13 - 1.23i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3 - 1.73i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.63 - 2.83i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.26 - 0.732i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.73 + 6.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.366 + 0.633i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.19iT - 31T^{2} \)
37 \( 1 + (3.46 + 2i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-7.56 - 4.36i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.86 + 3.23i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 10.1iT - 47T^{2} \)
53 \( 1 + 6.92T + 53T^{2} \)
59 \( 1 + (-9.29 + 5.36i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.23 - 2.13i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.79 - 2.76i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (8.02 - 4.63i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 5.39iT - 73T^{2} \)
79 \( 1 + 11.9T + 79T^{2} \)
83 \( 1 + 9.46iT - 83T^{2} \)
89 \( 1 + (4.09 + 2.36i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.25 + 4.76i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.42958942236308374829037298785, −10.12800715532802324687245436628, −8.869671923991677511897029653595, −8.038291688092229772009117252372, −7.01165603162752240332283685649, −6.22585695494828334158363293081, −5.52115330378584148318139057763, −4.48169602894667559041360027252, −3.61025246819828917785065051835, −2.47567692953724042443786491083, 1.39437661421867697270094453532, 2.92413227279555870393932829553, 3.59216314772988209217948646788, 4.55681093026786683565982298063, 5.66550140423748376240752630100, 6.63511954829206123865657179673, 7.11711262676284935733001755296, 9.220333173783665564293906093588, 9.836919201016920959490485689202, 10.86251718655384944997102170549

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