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Properties

Label 2-585-9.4-c1-0-26
Degree $2$
Conductor $585$
Sign $-0.608 - 0.793i$
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 + 2.23i)2-s + (1.68 − 0.410i)3-s + (−2.34 + 4.05i)4-s + (0.5 − 0.866i)5-s + (3.09 + 3.23i)6-s + (2.13 + 3.70i)7-s − 6.95·8-s + (2.66 − 1.38i)9-s + 2.58·10-s + (−2.19 − 3.79i)11-s + (−2.27 + 7.79i)12-s + (0.5 − 0.866i)13-s + (−5.52 + 9.57i)14-s + (0.485 − 1.66i)15-s + (−4.30 − 7.44i)16-s + 0.619·17-s + ⋯
L(s)  = 1  + (0.914 + 1.58i)2-s + (0.971 − 0.237i)3-s + (−1.17 + 2.02i)4-s + (0.223 − 0.387i)5-s + (1.26 + 1.32i)6-s + (0.807 + 1.39i)7-s − 2.45·8-s + (0.887 − 0.461i)9-s + 0.817·10-s + (−0.660 − 1.14i)11-s + (−0.656 + 2.25i)12-s + (0.138 − 0.240i)13-s + (−1.47 + 2.55i)14-s + (0.125 − 0.429i)15-s + (−1.07 − 1.86i)16-s + 0.150·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.608 - 0.793i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -0.608 - 0.793i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39757 + 2.83102i\)
\(L(\frac12)\) \(\approx\) \(1.39757 + 2.83102i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.68 + 0.410i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-1.29 - 2.23i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (-2.13 - 3.70i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.19 + 3.79i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 0.619T + 17T^{2} \)
19 \( 1 + 2.61T + 19T^{2} \)
23 \( 1 + (2.71 - 4.70i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.47 + 6.02i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.44 + 5.96i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.02T + 37T^{2} \)
41 \( 1 + (-2.03 + 3.52i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.01 - 8.69i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.202 - 0.351i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 8.71T + 53T^{2} \)
59 \( 1 + (-5.36 + 9.28i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.53 + 4.39i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.27 - 12.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.57T + 71T^{2} \)
73 \( 1 - 16.7T + 73T^{2} \)
79 \( 1 + (7.27 + 12.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.52 - 2.63i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 7.27T + 89T^{2} \)
97 \( 1 + (-4.81 - 8.34i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.31886572347399239218748894305, −9.570991272673117658968790406874, −8.745393359949861769701093240836, −8.082287332158941770898090707486, −7.74779374475475910112476311790, −6.19360202936910615211435500922, −5.70660877048268935330971913030, −4.73396413501297037763951827523, −3.56558073931073743370199406069, −2.33309427357688076210928844420, 1.53038100398638208290171981775, 2.41690708252113713300327897863, 3.61446248892015833869611747680, 4.39879083096349229176853171702, 5.04225234754303535022164173728, 6.83290285991297365402386803803, 7.79880810131217958377694623704, 8.987389546127545649230491402004, 10.07158092156799421311724320660, 10.48209180088947500895768467778

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