Properties

Label 2-585-65.29-c1-0-9
Degree $2$
Conductor $585$
Sign $0.547 - 0.836i$
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  + (1.16 − 0.672i)2-s + (−0.0962 + 0.166i)4-s + (0.868 + 2.06i)5-s + (−3.39 − 1.96i)7-s + 2.94i·8-s + (2.39 + 1.81i)10-s + (1.37 + 2.38i)11-s + (1.14 + 3.41i)13-s − 5.27·14-s + (1.78 + 3.09i)16-s + (4.09 + 2.36i)17-s + (1.85 − 3.20i)19-s + (−0.427 − 0.0536i)20-s + (3.20 + 1.84i)22-s + (−4.57 + 2.64i)23-s + ⋯
L(s)  = 1  + (0.823 − 0.475i)2-s + (−0.0481 + 0.0833i)4-s + (0.388 + 0.921i)5-s + (−1.28 − 0.741i)7-s + 1.04i·8-s + (0.757 + 0.574i)10-s + (0.414 + 0.718i)11-s + (0.318 + 0.947i)13-s − 1.40·14-s + (0.447 + 0.774i)16-s + (0.992 + 0.573i)17-s + (0.424 − 0.735i)19-s + (−0.0955 − 0.0119i)20-s + (0.683 + 0.394i)22-s + (−0.953 + 0.550i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.547 - 0.836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.547 - 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.547 - 0.836i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ 0.547 - 0.836i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67679 + 0.906219i\)
\(L(\frac12)\) \(\approx\) \(1.67679 + 0.906219i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-0.868 - 2.06i)T \)
13 \( 1 + (-1.14 - 3.41i)T \)
good2 \( 1 + (-1.16 + 0.672i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (3.39 + 1.96i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.37 - 2.38i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-4.09 - 2.36i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.85 + 3.20i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.57 - 2.64i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.31 - 2.27i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.71T + 31T^{2} \)
37 \( 1 + (3.66 - 2.11i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.408 - 0.708i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.85 + 2.80i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 8.36iT - 47T^{2} \)
53 \( 1 + 7.01iT - 53T^{2} \)
59 \( 1 + (1.09 - 1.89i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.41 + 11.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.8 + 6.28i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.08 + 10.5i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 0.955iT - 73T^{2} \)
79 \( 1 - 11.1T + 79T^{2} \)
83 \( 1 - 11.7iT - 83T^{2} \)
89 \( 1 + (-5.60 - 9.70i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.52 - 0.882i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.92279478916244729078415271687, −9.995475542051902113160401698968, −9.475985697708077090528636173955, −8.095570303338025643185953230192, −6.92767073811986488044097391015, −6.42341312264200061410844693721, −5.14386955690134779193149288520, −3.78890921874105593535018941610, −3.40340763134690460815730813668, −2.03423229776742785247903771917, 0.863422828009813118834504242209, 2.98400283177605228232531565170, 3.98381830389147740610943256632, 5.30843369226159918633240159052, 5.86236815381081192275202037397, 6.43718540103397948428179795630, 7.914894537292257753403882890984, 8.872886972016935644498383730596, 9.769060783763649989003833352229, 10.17928053180761439930412759501

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