Properties

Label 2-85-85.4-c1-0-6
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: $\chi_{85} (4, \cdot )$
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

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

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