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Properties

Label 2-585-117.25-c1-0-10
Degree $2$
Conductor $585$
Sign $0.754 + 0.656i$
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  + (−0.527 − 0.304i)2-s + (−1.60 − 0.654i)3-s + (−0.814 − 1.41i)4-s + (0.866 − 0.5i)5-s + (0.645 + 0.833i)6-s + (1.11 + 0.646i)7-s + 2.20i·8-s + (2.14 + 2.10i)9-s − 0.608·10-s + (2.32 + 1.34i)11-s + (0.382 + 2.79i)12-s + (2.47 + 2.62i)13-s + (−0.393 − 0.681i)14-s + (−1.71 + 0.234i)15-s + (−0.957 + 1.65i)16-s − 1.63·17-s + ⋯
L(s)  = 1  + (−0.372 − 0.215i)2-s + (−0.925 − 0.378i)3-s + (−0.407 − 0.705i)4-s + (0.387 − 0.223i)5-s + (0.263 + 0.340i)6-s + (0.423 + 0.244i)7-s + 0.780i·8-s + (0.714 + 0.700i)9-s − 0.192·10-s + (0.700 + 0.404i)11-s + (0.110 + 0.807i)12-s + (0.686 + 0.726i)13-s + (−0.105 − 0.182i)14-s + (−0.443 + 0.0605i)15-s + (−0.239 + 0.414i)16-s − 0.396·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.754 + 0.656i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.754 + 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.754 + 0.656i$
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: \((2,\ 585,\ (\ :1/2),\ 0.754 + 0.656i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.891395 - 0.333824i\)
\(L(\frac12)\) \(\approx\) \(0.891395 - 0.333824i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.60 + 0.654i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (-2.47 - 2.62i)T \)
good2 \( 1 + (0.527 + 0.304i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (-1.11 - 0.646i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.32 - 1.34i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 1.63T + 17T^{2} \)
19 \( 1 + 2.80iT - 19T^{2} \)
23 \( 1 + (-4.00 - 6.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.93 + 3.35i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-7.48 + 4.31i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 10.1iT - 37T^{2} \)
41 \( 1 + (7.08 - 4.09i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.05 - 10.4i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-10.5 - 6.07i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 3.10T + 53T^{2} \)
59 \( 1 + (-8.90 + 5.14i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.42 + 4.20i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (11.3 - 6.57i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.938iT - 71T^{2} \)
73 \( 1 + 14.6iT - 73T^{2} \)
79 \( 1 + (-3.27 + 5.67i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.19 - 5.30i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 10.4iT - 89T^{2} \)
97 \( 1 + (-9.03 - 5.21i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.71135058528840557252446109229, −9.650905138295010618081849981037, −9.137032030150852267489387739054, −8.048554257045450977273645150333, −6.77464951313292672837277061218, −6.04380389208705520948020761571, −5.09524589590095972373369591587, −4.32202846325502867149659083747, −2.03516360025237328701692768970, −1.06755091142168028157092568740, 0.977916120882256849868789322782, 3.23568293310638052737673929458, 4.26097324394129962227800088419, 5.24030110438166011980747460792, 6.45641079255300809609432457339, 7.01125877820351459677633435769, 8.447180263344594745191694149876, 8.825452571713737206828393331859, 10.24740003029942716410824128769, 10.47192042098995015276331427915

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