Properties

Label 2-585-9.7-c1-0-19
Degree $2$
Conductor $585$
Sign $-0.585 - 0.810i$
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  + (−0.247 + 0.427i)2-s + (0.674 + 1.59i)3-s + (0.877 + 1.52i)4-s + (0.5 + 0.866i)5-s + (−0.849 − 0.105i)6-s + (2.14 − 3.71i)7-s − 1.85·8-s + (−2.08 + 2.15i)9-s − 0.494·10-s + (−2.24 + 3.88i)11-s + (−1.83 + 2.42i)12-s + (0.5 + 0.866i)13-s + (1.05 + 1.83i)14-s + (−1.04 + 1.38i)15-s + (−1.29 + 2.24i)16-s + 1.57·17-s + ⋯
L(s)  = 1  + (−0.174 + 0.302i)2-s + (0.389 + 0.920i)3-s + (0.438 + 0.760i)4-s + (0.223 + 0.387i)5-s + (−0.346 − 0.0429i)6-s + (0.809 − 1.40i)7-s − 0.656·8-s + (−0.696 + 0.717i)9-s − 0.156·10-s + (−0.676 + 1.17i)11-s + (−0.529 + 0.700i)12-s + (0.138 + 0.240i)13-s + (0.282 + 0.490i)14-s + (−0.269 + 0.356i)15-s + (−0.324 + 0.561i)16-s + 0.381·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.585 - 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.585 - 0.810i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (196, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -0.585 - 0.810i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.779695 + 1.52588i\)
\(L(\frac12)\) \(\approx\) \(0.779695 + 1.52588i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.674 - 1.59i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (0.247 - 0.427i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (-2.14 + 3.71i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.24 - 3.88i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 1.57T + 17T^{2} \)
19 \( 1 - 1.86T + 19T^{2} \)
23 \( 1 + (-2.62 - 4.55i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.375 + 0.649i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.10 + 8.83i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2.78T + 37T^{2} \)
41 \( 1 + (-3.18 - 5.50i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.79 - 6.57i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.13 + 7.16i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 0.0752T + 53T^{2} \)
59 \( 1 + (5.72 + 9.90i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.98 + 10.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.44 - 11.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 14.4T + 71T^{2} \)
73 \( 1 - 9.37T + 73T^{2} \)
79 \( 1 + (-3.04 + 5.28i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.306 - 0.530i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 16.2T + 89T^{2} \)
97 \( 1 + (-7.49 + 12.9i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.14073958619367933424174022933, −9.984559061250960445029520947875, −9.465243733500902319851020119109, −7.995870314657939617479815599487, −7.70751090862240872363284596297, −6.82359514078878345445040972874, −5.32900669539309693430387823916, −4.28686029548123438850418176923, −3.44195170802660776608221335015, −2.13506426372550140385484782914, 1.01480494893843027980633457920, 2.21421014198268510835319625559, 3.03519936334191694044787628538, 5.36652077691987264875481675654, 5.60368913535721248415210153509, 6.71240945706453184568488134645, 7.962167416693124519131813617888, 8.722306935737297664629677258586, 9.215038063083320428599662474152, 10.62029970197243545353925578135

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