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Properties

Label 2-585-117.25-c1-0-0
Degree $2$
Conductor $585$
Sign $-0.854 - 0.519i$
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  + (−1.62 − 0.939i)2-s + (−1.72 + 0.161i)3-s + (0.764 + 1.32i)4-s + (−0.866 + 0.5i)5-s + (2.95 + 1.35i)6-s + (1.80 + 1.04i)7-s + 0.885i·8-s + (2.94 − 0.555i)9-s + 1.87·10-s + (−4.07 − 2.35i)11-s + (−1.53 − 2.15i)12-s + (−0.125 + 3.60i)13-s + (−1.95 − 3.39i)14-s + (1.41 − 1.00i)15-s + (2.36 − 4.08i)16-s + 4.67·17-s + ⋯
L(s)  = 1  + (−1.15 − 0.664i)2-s + (−0.995 + 0.0929i)3-s + (0.382 + 0.661i)4-s + (−0.387 + 0.223i)5-s + (1.20 + 0.554i)6-s + (0.682 + 0.394i)7-s + 0.313i·8-s + (0.982 − 0.185i)9-s + 0.594·10-s + (−1.22 − 0.708i)11-s + (−0.442 − 0.623i)12-s + (−0.0346 + 0.999i)13-s + (−0.523 − 0.906i)14-s + (0.364 − 0.258i)15-s + (0.590 − 1.02i)16-s + 1.13·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.854 - 0.519i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.854 - 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.854 - 0.519i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (376, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -0.854 - 0.519i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00144015 + 0.00513777i\)
\(L(\frac12)\) \(\approx\) \(0.00144015 + 0.00513777i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 - 0.161i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (0.125 - 3.60i)T \)
good2 \( 1 + (1.62 + 0.939i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (-1.80 - 1.04i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.07 + 2.35i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 4.67T + 17T^{2} \)
19 \( 1 + 4.04iT - 19T^{2} \)
23 \( 1 + (2.18 + 3.78i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.28 - 2.22i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.329 - 0.190i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.79iT - 37T^{2} \)
41 \( 1 + (10.4 - 6.03i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.00 - 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.347 - 0.200i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 12.5T + 53T^{2} \)
59 \( 1 + (-6.53 + 3.77i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.68 - 2.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.00 - 5.20i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.42iT - 71T^{2} \)
73 \( 1 + 1.66iT - 73T^{2} \)
79 \( 1 + (0.531 - 0.920i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.57 - 0.909i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 7.99iT - 89T^{2} \)
97 \( 1 + (14.4 + 8.34i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.01044753074661332468850652306, −10.33482378814293542675761999041, −9.548663215167989054301940964692, −8.441472965458740093865383571381, −7.84056683665706598512645510821, −6.69804681575525754210107790725, −5.45031251814556836528824391642, −4.71293532138132648471683617218, −2.98580570872141287290767950087, −1.55500520054538879250604634220, 0.00540738382213510135980793685, 1.46710648246933078752584199307, 3.76846404681735011929172372010, 5.09838641143529996355328327233, 5.78571378646381147399134460449, 7.12757556458866557745114364344, 7.82490179733555169223884431925, 8.084673620668245163086531707097, 9.649520402679120764155311198775, 10.26851642462212134844871877071

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