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Properties

Label 2-85-17.15-c1-0-4
Degree $2$
Conductor $85$
Sign $0.711 + 0.702i$
Analytic cond. $0.678728$
Root an. cond. $0.823849$
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  + (0.680 + 0.680i)2-s + (−1.01 − 2.44i)3-s − 1.07i·4-s + (−0.923 + 0.382i)5-s + (0.976 − 2.35i)6-s + (2.85 + 1.18i)7-s + (2.09 − 2.09i)8-s + (−2.84 + 2.84i)9-s + (−0.889 − 0.368i)10-s + (−2.34 + 5.66i)11-s + (−2.62 + 1.08i)12-s − 1.16i·13-s + (1.14 + 2.75i)14-s + (1.87 + 1.87i)15-s + 0.703·16-s + (1.25 − 3.92i)17-s + ⋯
L(s)  = 1  + (0.481 + 0.481i)2-s + (−0.585 − 1.41i)3-s − 0.536i·4-s + (−0.413 + 0.171i)5-s + (0.398 − 0.962i)6-s + (1.08 + 0.447i)7-s + (0.739 − 0.739i)8-s + (−0.946 + 0.946i)9-s + (−0.281 − 0.116i)10-s + (−0.707 + 1.70i)11-s + (−0.757 + 0.313i)12-s − 0.321i·13-s + (0.304 + 0.735i)14-s + (0.483 + 0.483i)15-s + 0.175·16-s + (0.305 − 0.952i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $0.711 + 0.702i$
Analytic conductor: \(0.678728\)
Root analytic conductor: \(0.823849\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (66, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 85,\ (\ :1/2),\ 0.711 + 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.967845 - 0.397114i\)
\(L(\frac12)\) \(\approx\) \(0.967845 - 0.397114i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.923 - 0.382i)T \)
17 \( 1 + (-1.25 + 3.92i)T \)
good2 \( 1 + (-0.680 - 0.680i)T + 2iT^{2} \)
3 \( 1 + (1.01 + 2.44i)T + (-2.12 + 2.12i)T^{2} \)
7 \( 1 + (-2.85 - 1.18i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (2.34 - 5.66i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + 1.16iT - 13T^{2} \)
19 \( 1 + (-3.83 - 3.83i)T + 19iT^{2} \)
23 \( 1 + (1.19 - 2.88i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (4.61 - 1.91i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (1.42 + 3.44i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (0.151 + 0.366i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-1.57 - 0.651i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (-0.0189 + 0.0189i)T - 43iT^{2} \)
47 \( 1 - 5.43iT - 47T^{2} \)
53 \( 1 + (0.244 + 0.244i)T + 53iT^{2} \)
59 \( 1 + (-2.87 + 2.87i)T - 59iT^{2} \)
61 \( 1 + (11.4 + 4.76i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + 5.62T + 67T^{2} \)
71 \( 1 + (-4.12 - 9.95i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (1.52 - 0.633i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (-2.01 + 4.87i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (8.78 + 8.78i)T + 83iT^{2} \)
89 \( 1 + 3.22iT - 89T^{2} \)
97 \( 1 + (-11.9 + 4.94i)T + (68.5 - 68.5i)T^{2} \)
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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

−14.17121348883406447198866836789, −13.02288874751909755927698994809, −12.14577537482450351078195706841, −11.24435158719900455481647427100, −9.822738169103701591085694778603, −7.61134984261568230546918830825, −7.38813230527674191610430998057, −5.79730450156628542030604841184, −4.88691117773629465018555276169, −1.75996424768553766619335775501, 3.42465488065181914391997182381, 4.48111923211612207638936495720, 5.48127203637532832781796205064, 7.81049110404248174468690449369, 8.811215207407080942777835111779, 10.61312727147919754163920145927, 11.08583476200567583409284123998, 11.85632119275467760944128780948, 13.34387287413421838893300974090, 14.31975840917048058449574331719

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