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Properties

Label 2-85-85.64-c1-0-2
Degree $2$
Conductor $85$
Sign $0.841 - 0.540i$
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  + 2.38·2-s + (−2.23 + 2.23i)3-s + 3.69·4-s + (−0.518 − 2.17i)5-s + (−5.32 + 5.32i)6-s + (−0.155 − 0.155i)7-s + 4.04·8-s − 6.95i·9-s + (−1.23 − 5.19i)10-s + (0.371 − 0.371i)11-s + (−8.24 + 8.24i)12-s + 1.96i·13-s + (−0.371 − 0.371i)14-s + (6.00 + 3.69i)15-s + 2.25·16-s + (−3.46 − 2.23i)17-s + ⋯
L(s)  = 1  + 1.68·2-s + (−1.28 + 1.28i)3-s + 1.84·4-s + (−0.232 − 0.972i)5-s + (−2.17 + 2.17i)6-s + (−0.0587 − 0.0587i)7-s + 1.42·8-s − 2.31i·9-s + (−0.391 − 1.64i)10-s + (0.111 − 0.111i)11-s + (−2.37 + 2.37i)12-s + 0.545i·13-s + (−0.0992 − 0.0992i)14-s + (1.55 + 0.953i)15-s + 0.564·16-s + (−0.841 − 0.541i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $0.841 - 0.540i$
Analytic conductor: \(0.678728\)
Root analytic conductor: \(0.823849\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 85,\ (\ :1/2),\ 0.841 - 0.540i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44837 + 0.424717i\)
\(L(\frac12)\) \(\approx\) \(1.44837 + 0.424717i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.518 + 2.17i)T \)
17 \( 1 + (3.46 + 2.23i)T \)
good2 \( 1 - 2.38T + 2T^{2} \)
3 \( 1 + (2.23 - 2.23i)T - 3iT^{2} \)
7 \( 1 + (0.155 + 0.155i)T + 7iT^{2} \)
11 \( 1 + (-0.371 + 0.371i)T - 11iT^{2} \)
13 \( 1 - 1.96iT - 13T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + (-0.263 - 0.263i)T + 23iT^{2} \)
29 \( 1 + (-4.95 - 4.95i)T + 29iT^{2} \)
31 \( 1 + (-2.06 - 2.06i)T + 31iT^{2} \)
37 \( 1 + (-4.04 + 4.04i)T - 37iT^{2} \)
41 \( 1 + (-0.563 + 0.563i)T - 41iT^{2} \)
43 \( 1 + 2.49T + 43T^{2} \)
47 \( 1 + 6.73iT - 47T^{2} \)
53 \( 1 - 5.92T + 53T^{2} \)
59 \( 1 + 6iT - 59T^{2} \)
61 \( 1 + (4 - 4i)T - 61iT^{2} \)
67 \( 1 - 11.5iT - 67T^{2} \)
71 \( 1 + (5.06 + 5.06i)T + 71iT^{2} \)
73 \( 1 + (-0.838 + 0.838i)T - 73iT^{2} \)
79 \( 1 + (4.75 - 4.75i)T - 79iT^{2} \)
83 \( 1 - 6.11T + 83T^{2} \)
89 \( 1 - 15.9T + 89T^{2} \)
97 \( 1 + (-6.51 + 6.51i)T - 97iT^{2} \)
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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

−14.42262852211416475189248640363, −13.14867601690493895608734304298, −12.03912219090028198252717258106, −11.60868506243728730399322720736, −10.42130518366568999782734453347, −9.034658500923577641857042605126, −6.64336922245928535451682241002, −5.49599665280826668564959823871, −4.68792302605709513863214319807, −3.81941776759537386019486819104, 2.57186858713637706785766573019, 4.61799291209982817122434556990, 6.06606416541646083383873448663, 6.57717039434460590148109472270, 7.67298757410771910104002976277, 10.64632810786923183878781233143, 11.41483035519605213052052709623, 12.14048152169788997512864278937, 13.12935816083549445613019059283, 13.73015016435936456578953292055

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