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Properties

Label 2-585-13.10-c1-0-5
Degree $2$
Conductor $585$
Sign $-0.0791 - 0.996i$
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  + (−1.29 + 0.747i)2-s + (0.118 − 0.204i)4-s − i·5-s + (−4.18 − 2.41i)7-s − 2.63i·8-s + (0.747 + 1.29i)10-s + (0.926 − 0.534i)11-s + (0.331 + 3.59i)13-s + 7.21·14-s + (2.20 + 3.82i)16-s + (−1.77 + 3.08i)17-s + (4.96 + 2.86i)19-s + (−0.204 − 0.118i)20-s + (−0.799 + 1.38i)22-s + (3.54 + 6.13i)23-s + ⋯
L(s)  = 1  + (−0.915 + 0.528i)2-s + (0.0591 − 0.102i)4-s − 0.447i·5-s + (−1.57 − 0.912i)7-s − 0.932i·8-s + (0.236 + 0.409i)10-s + (0.279 − 0.161i)11-s + (0.0918 + 0.995i)13-s + 1.92·14-s + (0.552 + 0.956i)16-s + (−0.431 + 0.747i)17-s + (1.13 + 0.657i)19-s + (−0.0458 − 0.0264i)20-s + (−0.170 + 0.295i)22-s + (0.738 + 1.27i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0791 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0791 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.0791 - 0.996i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -0.0791 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.377554 + 0.408706i\)
\(L(\frac12)\) \(\approx\) \(0.377554 + 0.408706i\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
13 \( 1 + (-0.331 - 3.59i)T \)
good2 \( 1 + (1.29 - 0.747i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (4.18 + 2.41i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.926 + 0.534i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.77 - 3.08i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.96 - 2.86i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.54 - 6.13i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.736 - 1.27i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.46iT - 31T^{2} \)
37 \( 1 + (0.0219 - 0.0126i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.232 + 0.133i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.77 + 3.08i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 6.51iT - 47T^{2} \)
53 \( 1 + 0.991T + 53T^{2} \)
59 \( 1 + (-7.55 - 4.36i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.16 - 5.48i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.48 - 2.58i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.72 - 3.88i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 10.1iT - 73T^{2} \)
79 \( 1 - 8.78T + 79T^{2} \)
83 \( 1 + 0.725iT - 83T^{2} \)
89 \( 1 + (11.6 - 6.75i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.97 + 1.71i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.55548559452901491406816626928, −9.699225877242514651862698031844, −9.280882050891246185369968102000, −8.399307805216956084477784901024, −7.22571152334745818022516173754, −6.83272029632000892803359438388, −5.76211935693528320534137809776, −4.07913668123172208420083581093, −3.43010500714002175036193578080, −1.13585218698366801377739690055, 0.50982224875709423994697132729, 2.54575940764480281079593014839, 3.13019511639156541836219237192, 4.99447724306036562433896379932, 6.02199659092725803318745653811, 6.89958304849212400260996847239, 8.058473425464262052945863164299, 9.194623269228530187602184290075, 9.441633458089639610586145942165, 10.32542587436769251292312545029

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