Properties

Label 2-585-9.7-c1-0-20
Degree $2$
Conductor $585$
Sign $-0.0542 - 0.998i$
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  + (−0.264 + 0.457i)2-s + (1.72 − 0.197i)3-s + (0.860 + 1.48i)4-s + (0.5 + 0.866i)5-s + (−0.364 + 0.840i)6-s + (−1.64 + 2.85i)7-s − 1.96·8-s + (2.92 − 0.680i)9-s − 0.528·10-s + (0.276 − 0.478i)11-s + (1.77 + 2.39i)12-s + (−0.5 − 0.866i)13-s + (−0.872 − 1.51i)14-s + (1.03 + 1.39i)15-s + (−1.20 + 2.07i)16-s + 6.44·17-s + ⋯
L(s)  = 1  + (−0.186 + 0.323i)2-s + (0.993 − 0.114i)3-s + (0.430 + 0.744i)4-s + (0.223 + 0.387i)5-s + (−0.148 + 0.343i)6-s + (−0.623 + 1.07i)7-s − 0.695·8-s + (0.973 − 0.226i)9-s − 0.167·10-s + (0.0832 − 0.144i)11-s + (0.512 + 0.690i)12-s + (−0.138 − 0.240i)13-s + (−0.233 − 0.403i)14-s + (0.266 + 0.359i)15-s + (−0.300 + 0.519i)16-s + 1.56·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0542 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0542 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.0542 - 0.998i$
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: \((2,\ 585,\ (\ :1/2),\ -0.0542 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33608 + 1.41064i\)
\(L(\frac12)\) \(\approx\) \(1.33608 + 1.41064i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.72 + 0.197i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.264 - 0.457i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (1.64 - 2.85i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.276 + 0.478i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 6.44T + 17T^{2} \)
19 \( 1 + 7.62T + 19T^{2} \)
23 \( 1 + (-1.05 - 1.83i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.658 - 1.14i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.07 + 7.06i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 8.74T + 37T^{2} \)
41 \( 1 + (-4.67 - 8.09i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.37 + 5.84i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.84 + 4.93i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 4.88T + 53T^{2} \)
59 \( 1 + (0.860 + 1.49i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.714 - 1.23i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.567 + 0.982i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.65T + 71T^{2} \)
73 \( 1 - 3.42T + 73T^{2} \)
79 \( 1 + (5.38 - 9.32i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.45 + 9.45i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 8.64T + 89T^{2} \)
97 \( 1 + (3.18 - 5.52i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.86732444493105783598886128589, −9.694920887311162825178980488672, −9.121114456150103553715317022549, −8.180908318622159887811497994219, −7.58372745217746141289944170675, −6.51537165304422964580497184912, −5.78061150576778027569684528985, −3.93867467272521802583227608184, −2.97873890976396981161998809499, −2.23251806280280058082418327306, 1.09588105869568455336104962579, 2.37615795268384936509687222742, 3.59823058228249351367276210135, 4.64682436327419848810718148883, 6.04111479697296424442774668968, 6.96150270975161771210239608403, 7.84595959187783899257245415644, 9.007622562534178897005073253909, 9.653278104532633994548626834835, 10.36975251128535074667584991484

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