Properties

Label 2-285-285.269-c1-0-27
Degree $2$
Conductor $285$
Sign $0.463 + 0.885i$
Analytic cond. $2.27573$
Root an. cond. $1.50855$
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  + (0.928 − 1.10i)2-s + (1.07 − 1.35i)3-s + (−0.0152 − 0.0867i)4-s + (0.0427 + 2.23i)5-s + (−0.508 − 2.45i)6-s + (1.83 − 1.06i)7-s + (2.39 + 1.38i)8-s + (−0.696 − 2.91i)9-s + (2.51 + 2.02i)10-s + (−3.86 − 2.23i)11-s + (−0.134 − 0.0722i)12-s + (0.689 + 0.250i)13-s + (0.532 − 3.01i)14-s + (3.08 + 2.34i)15-s + (3.91 − 1.42i)16-s + (−0.461 − 0.387i)17-s + ⋯
L(s)  = 1  + (0.656 − 0.782i)2-s + (0.619 − 0.784i)3-s + (−0.00764 − 0.0433i)4-s + (0.0191 + 0.999i)5-s + (−0.207 − 1.00i)6-s + (0.694 − 0.400i)7-s + (0.845 + 0.488i)8-s + (−0.232 − 0.972i)9-s + (0.795 + 0.641i)10-s + (−1.16 − 0.673i)11-s + (−0.0387 − 0.0208i)12-s + (0.191 + 0.0696i)13-s + (0.142 − 0.806i)14-s + (0.796 + 0.604i)15-s + (0.979 − 0.356i)16-s + (−0.111 − 0.0939i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.463 + 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.463 + 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $0.463 + 0.885i$
Analytic conductor: \(2.27573\)
Root analytic conductor: \(1.50855\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{285} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 285,\ (\ :1/2),\ 0.463 + 0.885i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.89734 - 1.14808i\)
\(L(\frac12)\) \(\approx\) \(1.89734 - 1.14808i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.07 + 1.35i)T \)
5 \( 1 + (-0.0427 - 2.23i)T \)
19 \( 1 + (3.57 - 2.48i)T \)
good2 \( 1 + (-0.928 + 1.10i)T + (-0.347 - 1.96i)T^{2} \)
7 \( 1 + (-1.83 + 1.06i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.86 + 2.23i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.689 - 0.250i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (0.461 + 0.387i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (-0.625 - 3.54i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-2.26 + 1.89i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (1.03 - 0.596i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 11.4T + 37T^{2} \)
41 \( 1 + (9.15 - 3.33i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-10.1 - 1.79i)T + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (-4.71 + 3.95i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (-9.54 + 1.68i)T + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (-6.54 - 5.49i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (1.77 + 10.0i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-4.62 + 3.88i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (1.01 - 5.78i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-0.863 - 2.37i)T + (-55.9 + 46.9i)T^{2} \)
79 \( 1 + (3.52 + 9.69i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (-4.00 - 6.93i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.94 + 2.16i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (8.69 + 7.29i)T + (16.8 + 95.5i)T^{2} \)
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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

−11.70905397032089335372786221455, −10.92765272036871641755501557107, −10.23034687619892942072570880173, −8.489017851285111134279800013133, −7.80179147801775633297037747929, −6.93061962079719216224515470219, −5.52228486741033745506701466041, −3.91845657140520846949806755639, −2.97024396576419581798622958304, −1.92324435807454003401316899236, 2.15110587153791109471198231131, 4.14394736649681782409535930120, 4.97161487663287318963750496039, 5.50253215154457495200004385180, 7.14589907479791435630575103723, 8.252344424817212763457120570822, 8.864777408688187647581514185416, 10.16335759586387091978821620612, 10.79094304642141070883363946297, 12.30403371122466550061601962638

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