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Properties

Label 2-585-65.64-c1-0-19
Degree $2$
Conductor $585$
Sign $0.165 + 0.986i$
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  − 0.246·2-s − 1.93·4-s + (2.15 + 0.608i)5-s − 2.59·7-s + 0.971·8-s + (−0.530 − 0.150i)10-s − 3.81i·11-s + (−1.54 − 3.25i)13-s + 0.639·14-s + 3.63·16-s + 4.63i·17-s − 5.02i·19-s + (−4.17 − 1.18i)20-s + 0.939i·22-s − 4i·23-s + ⋯
L(s)  = 1  − 0.174·2-s − 0.969·4-s + (0.962 + 0.272i)5-s − 0.979·7-s + 0.343·8-s + (−0.167 − 0.0474i)10-s − 1.14i·11-s + (−0.427 − 0.903i)13-s + 0.170·14-s + 0.909·16-s + 1.12i·17-s − 1.15i·19-s + (−0.932 − 0.263i)20-s + 0.200i·22-s − 0.834i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.165 + 0.986i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ 0.165 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.689135 - 0.583022i\)
\(L(\frac12)\) \(\approx\) \(0.689135 - 0.583022i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.15 - 0.608i)T \)
13 \( 1 + (1.54 + 3.25i)T \)
good2 \( 1 + 0.246T + 2T^{2} \)
7 \( 1 + 2.59T + 7T^{2} \)
11 \( 1 + 3.81iT - 11T^{2} \)
17 \( 1 - 4.63iT - 17T^{2} \)
19 \( 1 + 5.02iT - 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 - 7.87T + 29T^{2} \)
31 \( 1 + 10.2iT - 31T^{2} \)
37 \( 1 + 4.53T + 37T^{2} \)
41 \( 1 + 6.40iT - 41T^{2} \)
43 \( 1 - 0.639iT - 43T^{2} \)
47 \( 1 + 3.81T + 47T^{2} \)
53 \( 1 + 6.51iT - 53T^{2} \)
59 \( 1 + 6.24iT - 59T^{2} \)
61 \( 1 + 3.23T + 61T^{2} \)
67 \( 1 - 13.6T + 67T^{2} \)
71 \( 1 - 13.4iT - 71T^{2} \)
73 \( 1 + 3.08T + 73T^{2} \)
79 \( 1 + 3.36T + 79T^{2} \)
83 \( 1 - 7.23T + 83T^{2} \)
89 \( 1 - 8.83iT - 89T^{2} \)
97 \( 1 + 14.1T + 97T^{2} \)
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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

−10.24752993739743272811148158443, −9.738144974063899719594091272934, −8.803732303859333736656383067594, −8.147166180054963532554788253511, −6.70116505997491804222060240882, −5.94516953953538011863699323169, −5.05765598390454690636505841668, −3.67878459165844885797601210446, −2.63567476649855579623684330618, −0.57550655231728088314723537842, 1.54556426665050916525040912762, 3.10609098678954443671238728962, 4.52025081566075490061054999983, 5.20672189730495430273117988757, 6.41548340653443839132190412491, 7.24999863453090068020692429648, 8.524408979635239632657717735237, 9.463094620455335260454624770474, 9.731196050042089886216784716687, 10.43791406841880464070584169984

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