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Properties

Label 2-585-117.94-c1-0-24
Degree $2$
Conductor $585$
Sign $0.0363 - 0.999i$
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  + (−1.30 + 2.26i)2-s + (1.68 + 0.389i)3-s + (−2.40 − 4.17i)4-s + (0.5 − 0.866i)5-s + (−3.08 + 3.30i)6-s − 1.51·7-s + 7.36·8-s + (2.69 + 1.31i)9-s + (1.30 + 2.26i)10-s + (1.10 − 1.91i)11-s + (−2.44 − 7.98i)12-s + (1.72 + 3.16i)13-s + (1.98 − 3.43i)14-s + (1.18 − 1.26i)15-s + (−4.79 + 8.30i)16-s + (3.89 − 6.74i)17-s + ⋯
L(s)  = 1  + (−0.923 + 1.59i)2-s + (0.974 + 0.224i)3-s + (−1.20 − 2.08i)4-s + (0.223 − 0.387i)5-s + (−1.25 + 1.35i)6-s − 0.573·7-s + 2.60·8-s + (0.899 + 0.437i)9-s + (0.412 + 0.715i)10-s + (0.334 − 0.578i)11-s + (−0.705 − 2.30i)12-s + (0.477 + 0.878i)13-s + (0.529 − 0.917i)14-s + (0.304 − 0.327i)15-s + (−1.19 + 2.07i)16-s + (0.944 − 1.63i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0363 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0363 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.0363 - 0.999i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ 0.0363 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.913355 + 0.880700i\)
\(L(\frac12)\) \(\approx\) \(0.913355 + 0.880700i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.68 - 0.389i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-1.72 - 3.16i)T \)
good2 \( 1 + (1.30 - 2.26i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + 1.51T + 7T^{2} \)
11 \( 1 + (-1.10 + 1.91i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.89 + 6.74i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.35 + 2.34i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.82T + 23T^{2} \)
29 \( 1 + (2.73 - 4.74i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.21 - 5.56i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.67 - 6.35i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.650T + 41T^{2} \)
43 \( 1 - 4.61T + 43T^{2} \)
47 \( 1 + (5.92 + 10.2i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 10.3T + 53T^{2} \)
59 \( 1 + (-2.82 - 4.90i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 3.88T + 61T^{2} \)
67 \( 1 + 8.97T + 67T^{2} \)
71 \( 1 + (4.14 - 7.17i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 0.623T + 73T^{2} \)
79 \( 1 + (4.98 + 8.63i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.10 - 5.36i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.35 + 7.55i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.36T + 97T^{2} \)
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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

−10.29180190152476465922696828527, −9.482532951851645495003869134238, −9.021533719346298747225046212735, −8.466144387358355793494798370788, −7.22699224128535819186854743223, −6.89704159498613784312688036552, −5.58619632728505666474017787852, −4.72150102718869244400126507293, −3.21003143530044532908607144537, −1.16444272582000014359139758417, 1.25357386435126379012328761079, 2.37484750838401957704131478072, 3.40179662949223518661959807090, 3.98849894667941636387904777125, 6.04420736712490840693851956196, 7.52607946402852816243050388166, 8.044185514366776363232263675698, 9.078717059248183781238674604036, 9.709154756825151038682245123997, 10.30254440659145151049727791991

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