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Properties

Label 2-585-65.29-c1-0-28
Degree $2$
Conductor $585$
Sign $0.336 + 0.941i$
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  + (1.85 − 1.07i)2-s + (1.30 − 2.25i)4-s + (2.16 − 0.557i)5-s + (−0.635 − 0.367i)7-s − 1.30i·8-s + (3.42 − 3.36i)10-s + (0.110 + 0.190i)11-s + (0.632 − 3.54i)13-s − 1.57·14-s + (1.20 + 2.08i)16-s + (0.710 + 0.409i)17-s + (1.61 − 2.78i)19-s + (1.56 − 5.61i)20-s + (0.409 + 0.236i)22-s + (−6.92 + 3.99i)23-s + ⋯
L(s)  = 1  + (1.31 − 0.758i)2-s + (0.652 − 1.12i)4-s + (0.968 − 0.249i)5-s + (−0.240 − 0.138i)7-s − 0.461i·8-s + (1.08 − 1.06i)10-s + (0.0332 + 0.0575i)11-s + (0.175 − 0.984i)13-s − 0.421·14-s + (0.301 + 0.522i)16-s + (0.172 + 0.0994i)17-s + (0.369 − 0.639i)19-s + (0.350 − 1.25i)20-s + (0.0873 + 0.0504i)22-s + (−1.44 + 0.833i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.336 + 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.336 + 0.941i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ 0.336 + 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.67480 - 1.88379i\)
\(L(\frac12)\) \(\approx\) \(2.67480 - 1.88379i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-2.16 + 0.557i)T \)
13 \( 1 + (-0.632 + 3.54i)T \)
good2 \( 1 + (-1.85 + 1.07i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (0.635 + 0.367i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.110 - 0.190i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.710 - 0.409i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.61 + 2.78i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.92 - 3.99i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.51 - 2.62i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.27T + 31T^{2} \)
37 \( 1 + (4.44 - 2.56i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.87 - 10.1i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.62 + 2.66i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 5.80iT - 47T^{2} \)
53 \( 1 - 4.27iT - 53T^{2} \)
59 \( 1 + (1.08 - 1.88i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.03 - 10.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.38 + 0.797i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.41 - 7.65i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 7.86iT - 73T^{2} \)
79 \( 1 - 13.9T + 79T^{2} \)
83 \( 1 + 4.38iT - 83T^{2} \)
89 \( 1 + (2.16 + 3.74i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.48 - 4.32i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.56415201566302095354051465038, −10.04678173484994758707989473761, −8.987710682892511667140642781311, −7.82915565526231958941368245329, −6.45745810696442672664465533941, −5.61997692455357882168134590476, −4.97120233176218738333762086872, −3.72442920657325085630276607033, −2.77783825436078768549942466927, −1.55107656097371447258308168530, 2.05653298210160722344350855262, 3.42279309949247528176481217031, 4.42781117807525858260822511513, 5.53134794194381077344789201499, 6.17444067768157032916053709578, 6.85242725601660876236369212012, 7.909648592009412872332186525641, 9.203688598107425647866694277332, 9.926915694342753733164984920327, 10.96833456841578760175828762723

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