Properties

Label 2-585-117.25-c1-0-21
Degree $2$
Conductor $585$
Sign $-0.923 + 0.384i$
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  + (−2.39 − 1.38i)2-s + (−0.969 − 1.43i)3-s + (2.81 + 4.88i)4-s + (0.866 − 0.5i)5-s + (0.337 + 4.77i)6-s + (−0.814 − 0.470i)7-s − 10.0i·8-s + (−1.11 + 2.78i)9-s − 2.76·10-s + (1.21 + 0.701i)11-s + (4.27 − 8.77i)12-s + (−0.345 + 3.58i)13-s + (1.29 + 2.25i)14-s + (−1.55 − 0.758i)15-s + (−8.25 + 14.2i)16-s + 4.46·17-s + ⋯
L(s)  = 1  + (−1.69 − 0.977i)2-s + (−0.559 − 0.828i)3-s + (1.40 + 2.44i)4-s + (0.387 − 0.223i)5-s + (0.137 + 1.94i)6-s + (−0.307 − 0.177i)7-s − 3.55i·8-s + (−0.373 + 0.927i)9-s − 0.873·10-s + (0.366 + 0.211i)11-s + (1.23 − 2.53i)12-s + (−0.0957 + 0.995i)13-s + (0.347 + 0.601i)14-s + (−0.402 − 0.195i)15-s + (−2.06 + 3.57i)16-s + 1.08·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.384i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 + 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.923 + 0.384i$
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: \((2,\ 585,\ (\ :1/2),\ -0.923 + 0.384i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0925119 - 0.463208i\)
\(L(\frac12)\) \(\approx\) \(0.0925119 - 0.463208i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.969 + 1.43i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (0.345 - 3.58i)T \)
good2 \( 1 + (2.39 + 1.38i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (0.814 + 0.470i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.21 - 0.701i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 4.46T + 17T^{2} \)
19 \( 1 + 4.15iT - 19T^{2} \)
23 \( 1 + (3.91 + 6.78i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.28 + 7.42i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.20 + 3.58i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.11iT - 37T^{2} \)
41 \( 1 + (5.13 - 2.96i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.366 + 0.635i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.03 + 1.75i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 5.13T + 53T^{2} \)
59 \( 1 + (1.86 - 1.07i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.62 + 6.27i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.82 + 5.67i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.61iT - 71T^{2} \)
73 \( 1 - 2.38iT - 73T^{2} \)
79 \( 1 + (3.36 - 5.83i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (13.1 + 7.56i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 5.36iT - 89T^{2} \)
97 \( 1 + (2.13 + 1.23i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.01835759790113003725527428235, −9.805904132225415850349747807923, −8.525916810657801830257825508851, −8.003038818773014856131807874563, −6.81885030125591940321901698794, −6.39724026874209044969591163778, −4.43444047852105498782595624422, −2.78541853660275446952128099517, −1.79370100168172598886161792715, −0.56543546206931560650949148659, 1.24310737191129087526758227770, 3.21339946610604132478588854313, 5.26112702614128210656720200682, 5.81467751637667009665470366530, 6.59117447598992258091091561745, 7.68745519789371173501190842303, 8.534149198633635761415893008747, 9.448172003878725837761935175925, 10.10354558582734052297717440520, 10.45517578491812971347963452196

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