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Properties

Label 2-585-65.57-c1-0-32
Degree $2$
Conductor $585$
Sign $-0.994 + 0.100i$
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  + 0.792·2-s − 1.37·4-s + (−0.112 − 2.23i)5-s + 1.67i·7-s − 2.67·8-s + (−0.0891 − 1.77i)10-s + (−3.76 − 3.76i)11-s + (−2.58 + 2.51i)13-s + 1.32i·14-s + 0.623·16-s + (−3.99 + 3.99i)17-s + (0.167 + 0.167i)19-s + (0.154 + 3.06i)20-s + (−2.98 − 2.98i)22-s + (−0.0786 − 0.0786i)23-s + ⋯
L(s)  = 1  + 0.560·2-s − 0.685·4-s + (−0.0502 − 0.998i)5-s + 0.632i·7-s − 0.945·8-s + (−0.0281 − 0.559i)10-s + (−1.13 − 1.13i)11-s + (−0.715 + 0.698i)13-s + 0.354i·14-s + 0.155·16-s + (−0.968 + 0.968i)17-s + (0.0383 + 0.0383i)19-s + (0.0344 + 0.684i)20-s + (−0.636 − 0.636i)22-s + (−0.0163 − 0.0163i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.994 + 0.100i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -0.994 + 0.100i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0144147 - 0.287516i\)
\(L(\frac12)\) \(\approx\) \(0.0144147 - 0.287516i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.112 + 2.23i)T \)
13 \( 1 + (2.58 - 2.51i)T \)
good2 \( 1 - 0.792T + 2T^{2} \)
7 \( 1 - 1.67iT - 7T^{2} \)
11 \( 1 + (3.76 + 3.76i)T + 11iT^{2} \)
17 \( 1 + (3.99 - 3.99i)T - 17iT^{2} \)
19 \( 1 + (-0.167 - 0.167i)T + 19iT^{2} \)
23 \( 1 + (0.0786 + 0.0786i)T + 23iT^{2} \)
29 \( 1 + 9.61iT - 29T^{2} \)
31 \( 1 + (1.80 - 1.80i)T - 31iT^{2} \)
37 \( 1 + 5.41iT - 37T^{2} \)
41 \( 1 + (4.41 - 4.41i)T - 41iT^{2} \)
43 \( 1 + (9.01 + 9.01i)T + 43iT^{2} \)
47 \( 1 - 7.89iT - 47T^{2} \)
53 \( 1 + (0.968 - 0.968i)T - 53iT^{2} \)
59 \( 1 + (-3.71 + 3.71i)T - 59iT^{2} \)
61 \( 1 - 0.0803T + 61T^{2} \)
67 \( 1 - 1.98T + 67T^{2} \)
71 \( 1 + (1.90 - 1.90i)T - 71iT^{2} \)
73 \( 1 - 13.9T + 73T^{2} \)
79 \( 1 + 12.4iT - 79T^{2} \)
83 \( 1 + 4.05iT - 83T^{2} \)
89 \( 1 + (4.23 - 4.23i)T - 89iT^{2} \)
97 \( 1 + 3.53T + 97T^{2} \)
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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

−10.17794794085309332767263836943, −9.200867467683911261056085128146, −8.592309728889558409564867214316, −7.922343440444526348024290597256, −6.25142030105490386742617685435, −5.47086527126106785215948569197, −4.72429801524543199816369088816, −3.74223340045904546702911655896, −2.28758010050732775508870121780, −0.12782498109147572147066740541, 2.49404769010217335453370170198, 3.46622064120183317069115553027, 4.72202929844521928470709213055, 5.29439503565537717888481490072, 6.78639814782001745328262573228, 7.35179136807279475898844104171, 8.392955059400973495763264738757, 9.690770166526023123649479233640, 10.17009505463925580173111438351, 11.04603993305643369608445238048

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