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Properties

Label 2-585-65.29-c1-0-30
Degree $2$
Conductor $585$
Sign $-0.248 + 0.968i$
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  + (2.32 − 1.34i)2-s + (2.61 − 4.52i)4-s + (−2.23 − 0.0881i)5-s + (3.14 + 1.81i)7-s − 8.65i·8-s + (−5.31 + 2.79i)10-s + (−1.80 − 3.12i)11-s + (1.66 − 3.19i)13-s + 9.75·14-s + (−6.40 − 11.1i)16-s + (1.37 + 0.795i)17-s + (−2.37 + 4.11i)19-s + (−6.23 + 9.87i)20-s + (−8.40 − 4.85i)22-s + (−3.63 + 2.10i)23-s + ⋯
L(s)  = 1  + (1.64 − 0.950i)2-s + (1.30 − 2.26i)4-s + (−0.999 − 0.0394i)5-s + (1.18 + 0.685i)7-s − 3.06i·8-s + (−1.68 + 0.884i)10-s + (−0.544 − 0.943i)11-s + (0.461 − 0.887i)13-s + 2.60·14-s + (−1.60 − 2.77i)16-s + (0.334 + 0.192i)17-s + (−0.544 + 0.943i)19-s + (−1.39 + 2.20i)20-s + (−1.79 − 1.03i)22-s + (−0.758 + 0.438i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.248 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.248 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.248 + 0.968i$
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: \((2,\ 585,\ (\ :1/2),\ -0.248 + 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.09093 - 2.69371i\)
\(L(\frac12)\) \(\approx\) \(2.09093 - 2.69371i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 + 0.0881i)T \)
13 \( 1 + (-1.66 + 3.19i)T \)
good2 \( 1 + (-2.32 + 1.34i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (-3.14 - 1.81i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.80 + 3.12i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.37 - 0.795i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.37 - 4.11i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.63 - 2.10i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.32 - 4.02i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.07T + 31T^{2} \)
37 \( 1 + (7.72 - 4.46i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.49 - 6.05i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.42 + 0.823i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 6.71iT - 47T^{2} \)
53 \( 1 + 6.29iT - 53T^{2} \)
59 \( 1 + (-0.407 + 0.705i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.29 - 9.17i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.53 + 4.34i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.74 + 4.76i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 1.32iT - 73T^{2} \)
79 \( 1 + 6.22T + 79T^{2} \)
83 \( 1 + 6.63iT - 83T^{2} \)
89 \( 1 + (-6.68 - 11.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.41 - 3.12i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.83472866389770866717616655525, −10.20978907795757637076633307116, −8.462352842339831779584581789831, −7.909576168423807674242453380348, −6.30926046305667491629203751596, −5.47297827737127292398788378043, −4.73646168057618484254443676175, −3.66858696424215530117384270951, −2.84423147817661397552487489364, −1.36848296974889646142779315317, 2.35230668105208231704759433943, 3.88568689281105392301349598476, 4.46096086481400417590491452739, 5.08660675053256562375558520189, 6.50872725044028461297525195990, 7.23762260094329744909041275970, 7.87747345376336343097434894525, 8.621177852783627914482241758346, 10.50476645986087234274568472633, 11.36959525860027577577580923751

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