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Properties

Label 2-585-65.57-c1-0-12
Degree $2$
Conductor $585$
Sign $0.958 + 0.285i$
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  − 2.48·2-s + 4.18·4-s + (0.999 − 2.00i)5-s − 0.242i·7-s − 5.43·8-s + (−2.48 + 4.97i)10-s + (4.24 + 4.24i)11-s + (2.25 − 2.81i)13-s + 0.602i·14-s + 5.13·16-s + (−1.37 + 1.37i)17-s + (3.91 + 3.91i)19-s + (4.18 − 8.36i)20-s + (−10.5 − 10.5i)22-s + (1.90 + 1.90i)23-s + ⋯
L(s)  = 1  − 1.75·2-s + 2.09·4-s + (0.446 − 0.894i)5-s − 0.0916i·7-s − 1.92·8-s + (−0.785 + 1.57i)10-s + (1.27 + 1.27i)11-s + (0.624 − 0.781i)13-s + 0.161i·14-s + 1.28·16-s + (−0.332 + 0.332i)17-s + (0.898 + 0.898i)19-s + (0.934 − 1.87i)20-s + (−2.24 − 2.24i)22-s + (0.397 + 0.397i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.285i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 + 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.958 + 0.285i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ 0.958 + 0.285i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.773251 - 0.112819i\)
\(L(\frac12)\) \(\approx\) \(0.773251 - 0.112819i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.999 + 2.00i)T \)
13 \( 1 + (-2.25 + 2.81i)T \)
good2 \( 1 + 2.48T + 2T^{2} \)
7 \( 1 + 0.242iT - 7T^{2} \)
11 \( 1 + (-4.24 - 4.24i)T + 11iT^{2} \)
17 \( 1 + (1.37 - 1.37i)T - 17iT^{2} \)
19 \( 1 + (-3.91 - 3.91i)T + 19iT^{2} \)
23 \( 1 + (-1.90 - 1.90i)T + 23iT^{2} \)
29 \( 1 + 5.76iT - 29T^{2} \)
31 \( 1 + (5.69 - 5.69i)T - 31iT^{2} \)
37 \( 1 - 3.31iT - 37T^{2} \)
41 \( 1 + (-1.51 + 1.51i)T - 41iT^{2} \)
43 \( 1 + (3.88 + 3.88i)T + 43iT^{2} \)
47 \( 1 - 1.68iT - 47T^{2} \)
53 \( 1 + (-2.22 + 2.22i)T - 53iT^{2} \)
59 \( 1 + (-5.27 + 5.27i)T - 59iT^{2} \)
61 \( 1 - 10.2T + 61T^{2} \)
67 \( 1 - 15.3T + 67T^{2} \)
71 \( 1 + (-0.0780 + 0.0780i)T - 71iT^{2} \)
73 \( 1 + 1.45T + 73T^{2} \)
79 \( 1 + 7.60iT - 79T^{2} \)
83 \( 1 - 2.71iT - 83T^{2} \)
89 \( 1 + (-0.887 + 0.887i)T - 89iT^{2} \)
97 \( 1 + 4.76T + 97T^{2} \)
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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

−10.17815295751606331255105062225, −9.742214843557179342862078715278, −8.964158334804004033223554999390, −8.287175396814108116544667206317, −7.35343134457904489655451161204, −6.48032269967317828180983478056, −5.35528228771378839534288584426, −3.84220470564082289043655088610, −1.96566734144686640926525765752, −1.08997340395316174347850674924, 1.08085629536629432290678947922, 2.44041630065352548254723731471, 3.65586415256174642096228985002, 5.72230141974739296072490891548, 6.72883460630730129641065913301, 7.08292935825033782028069796823, 8.401672136873618842259462905711, 9.152738963605385172106462797430, 9.538645155255979637481254878235, 10.77294740204557094871281278377

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