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Properties

Label 2-585-117.25-c1-0-5
Degree $2$
Conductor $585$
Sign $0.0300 - 0.999i$
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  + (−2.03 − 1.17i)2-s + (0.447 + 1.67i)3-s + (1.75 + 3.03i)4-s + (0.866 − 0.5i)5-s + (1.05 − 3.92i)6-s + (−2.60 − 1.50i)7-s − 3.52i·8-s + (−2.59 + 1.49i)9-s − 2.34·10-s + (4.21 + 2.43i)11-s + (−4.29 + 4.29i)12-s + (−3.56 + 0.521i)13-s + (3.53 + 6.11i)14-s + (1.22 + 1.22i)15-s + (−0.635 + 1.10i)16-s + 3.52·17-s + ⋯
L(s)  = 1  + (−1.43 − 0.829i)2-s + (0.258 + 0.966i)3-s + (0.876 + 1.51i)4-s + (0.387 − 0.223i)5-s + (0.429 − 1.60i)6-s + (−0.985 − 0.569i)7-s − 1.24i·8-s + (−0.866 + 0.499i)9-s − 0.741·10-s + (1.27 + 0.733i)11-s + (−1.23 + 1.23i)12-s + (−0.989 + 0.144i)13-s + (0.943 + 1.63i)14-s + (0.316 + 0.316i)15-s + (−0.158 + 0.275i)16-s + 0.855·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0300 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0300 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.0300 - 0.999i$
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: \((2,\ 585,\ (\ :1/2),\ 0.0300 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.370761 + 0.359794i\)
\(L(\frac12)\) \(\approx\) \(0.370761 + 0.359794i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.447 - 1.67i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (3.56 - 0.521i)T \)
good2 \( 1 + (2.03 + 1.17i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (2.60 + 1.50i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.21 - 2.43i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 3.52T + 17T^{2} \)
19 \( 1 - 1.76iT - 19T^{2} \)
23 \( 1 + (-0.803 - 1.39i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.11 - 7.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.57 - 2.06i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.97iT - 37T^{2} \)
41 \( 1 + (1.46 - 0.843i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.47 - 4.29i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.99 + 1.14i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 0.0235T + 53T^{2} \)
59 \( 1 + (-12.0 + 6.98i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.41 - 12.8i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (12.2 - 7.04i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.4iT - 71T^{2} \)
73 \( 1 + 0.0924iT - 73T^{2} \)
79 \( 1 + (-4.65 + 8.05i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.07 + 2.35i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 16.2iT - 89T^{2} \)
97 \( 1 + (-11.3 - 6.56i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.36663369782027486649595444101, −9.988211579894296787096770898144, −9.389490322468715541629860347801, −8.849283971607871316334400566936, −7.63336784515372787601070653840, −6.76540643928102333911133152847, −5.24356442695128864008542150825, −3.87977278435089865229294819659, −2.98814601834934162416855580440, −1.54389416874504967323250600444, 0.45326505397202766935478181536, 2.00700214200044491662080631458, 3.34676816030305574606504746301, 5.74302383045897864552123030860, 6.23593183898468002062801062064, 7.05525536908585945329507114644, 7.75658144818255148746530766089, 8.835811815720940386310753203198, 9.313955231629606952342574537666, 9.983554408685508767115576908678

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