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Properties

Label 2-585-117.25-c1-0-12
Degree $2$
Conductor $585$
Sign $0.628 + 0.777i$
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  + (−1.07 − 0.619i)2-s + (−1.48 + 0.896i)3-s + (−0.232 − 0.403i)4-s + (0.866 − 0.5i)5-s + (2.14 − 0.0433i)6-s + (0.166 + 0.0962i)7-s + 3.05i·8-s + (1.39 − 2.65i)9-s − 1.23·10-s + (0.223 + 0.129i)11-s + (0.706 + 0.388i)12-s + (−3.45 + 1.03i)13-s + (−0.119 − 0.206i)14-s + (−0.835 + 1.51i)15-s + (1.42 − 2.47i)16-s + 3.42·17-s + ⋯
L(s)  = 1  + (−0.758 − 0.437i)2-s + (−0.855 + 0.517i)3-s + (−0.116 − 0.201i)4-s + (0.387 − 0.223i)5-s + (0.875 − 0.0177i)6-s + (0.0629 + 0.0363i)7-s + 1.07i·8-s + (0.464 − 0.885i)9-s − 0.391·10-s + (0.0674 + 0.0389i)11-s + (0.203 + 0.112i)12-s + (−0.958 + 0.286i)13-s + (−0.0318 − 0.0551i)14-s + (−0.215 + 0.391i)15-s + (0.356 − 0.617i)16-s + 0.830·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.628 + 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.628 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.628 + 0.777i$
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: \((2,\ 585,\ (\ :1/2),\ 0.628 + 0.777i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.607387 - 0.290065i\)
\(L(\frac12)\) \(\approx\) \(0.607387 - 0.290065i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.48 - 0.896i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (3.45 - 1.03i)T \)
good2 \( 1 + (1.07 + 0.619i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (-0.166 - 0.0962i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.223 - 0.129i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 3.42T + 17T^{2} \)
19 \( 1 - 2.08iT - 19T^{2} \)
23 \( 1 + (-0.614 - 1.06i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.55 + 7.89i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.13 - 1.23i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.41iT - 37T^{2} \)
41 \( 1 + (-9.38 + 5.41i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.21 + 9.02i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.32 + 1.34i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 9.93T + 53T^{2} \)
59 \( 1 + (-10.3 + 5.96i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.57 + 4.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.23 + 4.75i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.11iT - 71T^{2} \)
73 \( 1 - 10.7iT - 73T^{2} \)
79 \( 1 + (2.92 - 5.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.67 + 2.12i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 2.03iT - 89T^{2} \)
97 \( 1 + (5.71 + 3.30i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.26373244674613440092102111667, −9.919915522171734403334878942273, −9.201141490488225127572316708047, −8.181120874528797095511014825743, −6.97626580555519002610126842055, −5.72535136785355837701632604874, −5.19590440358081571453720850653, −4.03833465673396231525140769285, −2.23895466189484072459518739512, −0.74096035522849500307214772551, 1.00161407851067801108164716053, 2.80453365018707489386519406089, 4.45780279680910800427265638285, 5.52438840932148763830638617004, 6.55249372793616021150952043616, 7.31527950506970131936825495386, 7.941210204214446266017516271459, 9.081956755848301526642298585792, 9.935191053611517826089001901067, 10.64202571069523551763070380614

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