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Properties

Label 2-585-117.25-c1-0-50
Degree $2$
Conductor $585$
Sign $-0.641 + 0.767i$
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  + (0.231 + 0.133i)2-s + (1.70 − 0.313i)3-s + (−0.964 − 1.67i)4-s + (−0.866 + 0.5i)5-s + (0.436 + 0.155i)6-s + (−3.30 − 1.90i)7-s − 1.05i·8-s + (2.80 − 1.06i)9-s − 0.267·10-s + (−4.13 − 2.38i)11-s + (−2.16 − 2.54i)12-s + (3.37 + 1.27i)13-s + (−0.510 − 0.884i)14-s + (−1.31 + 1.12i)15-s + (−1.78 + 3.09i)16-s − 4.49·17-s + ⋯
L(s)  = 1  + (0.163 + 0.0945i)2-s + (0.983 − 0.180i)3-s + (−0.482 − 0.835i)4-s + (−0.387 + 0.223i)5-s + (0.178 + 0.0633i)6-s + (−1.25 − 0.721i)7-s − 0.371i·8-s + (0.934 − 0.355i)9-s − 0.0845·10-s + (−1.24 − 0.719i)11-s + (−0.625 − 0.734i)12-s + (0.935 + 0.354i)13-s + (−0.136 − 0.236i)14-s + (−0.340 + 0.289i)15-s + (−0.446 + 0.774i)16-s − 1.08·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.641 + 0.767i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.641 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.641 + 0.767i$
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: \((2,\ 585,\ (\ :1/2),\ -0.641 + 0.767i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.487000 - 1.04208i\)
\(L(\frac12)\) \(\approx\) \(0.487000 - 1.04208i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.70 + 0.313i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (-3.37 - 1.27i)T \)
good2 \( 1 + (-0.231 - 0.133i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (3.30 + 1.90i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.13 + 2.38i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 4.49T + 17T^{2} \)
19 \( 1 - 0.832iT - 19T^{2} \)
23 \( 1 + (2.95 + 5.11i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.72 + 2.98i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.11 + 2.37i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.42iT - 37T^{2} \)
41 \( 1 + (1.22 - 0.707i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.57 + 7.91i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-11.2 - 6.49i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 8.34T + 53T^{2} \)
59 \( 1 + (-0.991 + 0.572i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.08 - 8.81i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.5 + 6.08i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.12iT - 71T^{2} \)
73 \( 1 + 6.20iT - 73T^{2} \)
79 \( 1 + (-7.21 + 12.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.32 - 4.22i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 5.87iT - 89T^{2} \)
97 \( 1 + (12.3 + 7.13i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.45113690310715457706969302277, −9.456067627345003990030679637343, −8.689547720272661891286552676787, −7.80645233175630873438948286497, −6.64598457476382376493346340504, −6.07894478226650579674073770508, −4.43976712791513577269752537749, −3.70361900841438319629031846383, −2.50235108981030253941087013658, −0.53112915496715897744030148530, 2.46497247354842172013933723950, 3.24515309890458308058968546188, 4.17700972047524079031417197859, 5.24876487415168172016474079106, 6.70971199088942675225995403368, 7.75254077334507011775125084231, 8.419690535897490246016315932011, 9.162572554245959604780285328380, 9.865357415155112304988189567740, 10.95912572757438627479639716104

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