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Properties

Label 2-85-17.8-c1-0-1
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: $\chi_{85} (76, \cdot )$
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

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

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