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Properties

Label 2-585-117.25-c1-0-30
Degree $2$
Conductor $585$
Sign $-0.746 + 0.665i$
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  + (−1.71 − 0.988i)2-s + (1.03 + 1.38i)3-s + (0.952 + 1.65i)4-s + (−0.866 + 0.5i)5-s + (−0.410 − 3.39i)6-s + (−1.22 − 0.707i)7-s + 0.186i·8-s + (−0.837 + 2.88i)9-s + 1.97·10-s + (−2.00 − 1.15i)11-s + (−1.29 + 3.03i)12-s + (−0.495 − 3.57i)13-s + (1.39 + 2.42i)14-s + (−1.59 − 0.679i)15-s + (2.08 − 3.61i)16-s − 3.18·17-s + ⋯
L(s)  = 1  + (−1.21 − 0.698i)2-s + (0.600 + 0.799i)3-s + (0.476 + 0.825i)4-s + (−0.387 + 0.223i)5-s + (−0.167 − 1.38i)6-s + (−0.463 − 0.267i)7-s + 0.0658i·8-s + (−0.279 + 0.960i)9-s + 0.624·10-s + (−0.604 − 0.348i)11-s + (−0.373 + 0.876i)12-s + (−0.137 − 0.990i)13-s + (0.373 + 0.647i)14-s + (−0.411 − 0.175i)15-s + (0.522 − 0.904i)16-s − 0.772·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.746 + 0.665i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (376, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -0.746 + 0.665i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.114517 - 0.300773i\)
\(L(\frac12)\) \(\approx\) \(0.114517 - 0.300773i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.03 - 1.38i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (0.495 + 3.57i)T \)
good2 \( 1 + (1.71 + 0.988i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (1.22 + 0.707i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.00 + 1.15i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 3.18T + 17T^{2} \)
19 \( 1 + 0.980iT - 19T^{2} \)
23 \( 1 + (1.24 + 2.15i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.49 + 2.58i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.20 + 3.00i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 3.28iT - 37T^{2} \)
41 \( 1 + (-2.06 + 1.19i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.60 + 7.97i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.30 + 3.06i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 9.53T + 53T^{2} \)
59 \( 1 + (-1.50 + 0.871i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.11 - 3.65i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (11.5 - 6.69i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.496iT - 71T^{2} \)
73 \( 1 - 12.1iT - 73T^{2} \)
79 \( 1 + (2.20 - 3.81i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.14 + 3.55i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 9.63iT - 89T^{2} \)
97 \( 1 + (10.4 + 6.06i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.26190309771539130213113376572, −9.692714944279581073702614766744, −8.655411980455620515212176491962, −8.164147994522845107971688278451, −7.25888958534949568113180422118, −5.69556652833754012085156559875, −4.44185496157277760314985714166, −3.17500329914258083960811960847, −2.40273567450664355481963900213, −0.24734709009938473014647997992, 1.53679041738294416528333855185, 3.02619278336499111159816906206, 4.48789652233740178468300345700, 6.21087563239708844832784138129, 6.77854292088672878928955238973, 7.72261029299673077992105449753, 8.245533634761428165134838105507, 9.216439729129217527033058239754, 9.585670998378455638381941918920, 10.82993615117613688341791290467

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