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Properties

Label 2-585-65.9-c1-0-31
Degree $2$
Conductor $585$
Sign $-0.214 + 0.976i$
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  + (0.160 + 0.0926i)2-s + (−0.982 − 1.70i)4-s + (1.29 − 1.82i)5-s + (1.66 − 0.959i)7-s − 0.734i·8-s + (0.376 − 0.171i)10-s + (1.95 − 3.39i)11-s + (−2.82 + 2.24i)13-s + 0.355·14-s + (−1.89 + 3.28i)16-s + (−2.88 + 1.66i)17-s + (0.645 + 1.11i)19-s + (−4.37 − 0.420i)20-s + (0.628 − 0.362i)22-s + (−5.24 − 3.02i)23-s + ⋯
L(s)  = 1  + (0.113 + 0.0654i)2-s + (−0.491 − 0.851i)4-s + (0.580 − 0.814i)5-s + (0.628 − 0.362i)7-s − 0.259i·8-s + (0.119 − 0.0543i)10-s + (0.590 − 1.02i)11-s + (−0.782 + 0.622i)13-s + 0.0950·14-s + (−0.474 + 0.821i)16-s + (−0.699 + 0.403i)17-s + (0.148 + 0.256i)19-s + (−0.978 − 0.0941i)20-s + (0.133 − 0.0773i)22-s + (−1.09 − 0.630i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.214 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.214 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.214 + 0.976i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (334, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -0.214 + 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.920779 - 1.14491i\)
\(L(\frac12)\) \(\approx\) \(0.920779 - 1.14491i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.29 + 1.82i)T \)
13 \( 1 + (2.82 - 2.24i)T \)
good2 \( 1 + (-0.160 - 0.0926i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (-1.66 + 0.959i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.95 + 3.39i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.88 - 1.66i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.645 - 1.11i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.24 + 3.02i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.97 + 8.61i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.08T + 31T^{2} \)
37 \( 1 + (-8.81 - 5.09i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.79 + 4.84i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (7.62 - 4.40i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 9.87iT - 47T^{2} \)
53 \( 1 - 7.52iT - 53T^{2} \)
59 \( 1 + (-2.77 - 4.80i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.38 + 7.58i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.09 - 2.36i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.63 - 9.75i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 1.67iT - 73T^{2} \)
79 \( 1 - 0.965T + 79T^{2} \)
83 \( 1 + 11.4iT - 83T^{2} \)
89 \( 1 + (1.78 - 3.08i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.05 + 4.64i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.20679775456656217527750811150, −9.685634190617986427424295381323, −8.704587920124955768346975722214, −8.093901365127815662721041232649, −6.45696141999110354152700600563, −5.92971710153977340571443911294, −4.69877162563986329225538043424, −4.23210780043651846640710358630, −2.10681077300211761706556329378, −0.833104727639890712907442290232, 2.09795149028819100183290737136, 3.08788591348014071343032253295, 4.42684436203208410901817921215, 5.23355618684162031826393985749, 6.59080412508812148258828003727, 7.41267051481884875486522182695, 8.231980980300197765239987018278, 9.355714105073148494570107725785, 9.892582520321297474877487840707, 11.06206399637751774525874760260

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