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Properties

Label 2-85-17.13-c1-0-2
Degree $2$
Conductor $85$
Sign $0.978 + 0.204i$
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  + 0.677i·2-s + (−1.66 − 1.66i)3-s + 1.54·4-s + (0.707 + 0.707i)5-s + (1.12 − 1.12i)6-s + (3.02 − 3.02i)7-s + 2.39i·8-s + 2.55i·9-s + (−0.479 + 0.479i)10-s + (−1.17 + 1.17i)11-s + (−2.56 − 2.56i)12-s − 6.21·13-s + (2.04 + 2.04i)14-s − 2.35i·15-s + 1.45·16-s + (−1.32 + 3.90i)17-s + ⋯
L(s)  = 1  + 0.479i·2-s + (−0.962 − 0.962i)3-s + 0.770·4-s + (0.316 + 0.316i)5-s + (0.461 − 0.461i)6-s + (1.14 − 1.14i)7-s + 0.848i·8-s + 0.852i·9-s + (−0.151 + 0.151i)10-s + (−0.353 + 0.353i)11-s + (−0.741 − 0.741i)12-s − 1.72·13-s + (0.547 + 0.547i)14-s − 0.608i·15-s + 0.363·16-s + (−0.322 + 0.946i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.204i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 + 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $0.978 + 0.204i$
Analytic conductor: \(0.678728\)
Root analytic conductor: \(0.823849\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 85,\ (\ :1/2),\ 0.978 + 0.204i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.927921 - 0.0958634i\)
\(L(\frac12)\) \(\approx\) \(0.927921 - 0.0958634i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.707 - 0.707i)T \)
17 \( 1 + (1.32 - 3.90i)T \)
good2 \( 1 - 0.677iT - 2T^{2} \)
3 \( 1 + (1.66 + 1.66i)T + 3iT^{2} \)
7 \( 1 + (-3.02 + 3.02i)T - 7iT^{2} \)
11 \( 1 + (1.17 - 1.17i)T - 11iT^{2} \)
13 \( 1 + 6.21T + 13T^{2} \)
19 \( 1 - 3.38iT - 19T^{2} \)
23 \( 1 + (-3.30 + 3.30i)T - 23iT^{2} \)
29 \( 1 + (2.57 + 2.57i)T + 29iT^{2} \)
31 \( 1 + (2.12 + 2.12i)T + 31iT^{2} \)
37 \( 1 + (-3.78 - 3.78i)T + 37iT^{2} \)
41 \( 1 + (1.54 - 1.54i)T - 41iT^{2} \)
43 \( 1 + 0.998iT - 43T^{2} \)
47 \( 1 + 2.00T + 47T^{2} \)
53 \( 1 - 6.95iT - 53T^{2} \)
59 \( 1 + 6.30iT - 59T^{2} \)
61 \( 1 + (-4.62 + 4.62i)T - 61iT^{2} \)
67 \( 1 - 5.80T + 67T^{2} \)
71 \( 1 + (9.60 + 9.60i)T + 71iT^{2} \)
73 \( 1 + (7.01 + 7.01i)T + 73iT^{2} \)
79 \( 1 + (-0.820 + 0.820i)T - 79iT^{2} \)
83 \( 1 + 3.65iT - 83T^{2} \)
89 \( 1 - 2.69T + 89T^{2} \)
97 \( 1 + (-7.40 - 7.40i)T + 97iT^{2} \)
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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

−14.45672599017656841088426501471, −13.03129187931944634128179906330, −12.00089318273813542714913559178, −11.07858738193267332072024572602, −10.25436280304872658646108329932, −7.82348374944956424079512718635, −7.28518056513200426219263980195, −6.24568152743385341032883694510, −4.92168082018840481763794872167, −1.90901283877240005840362122233, 2.46116072350025363969141529904, 4.88492290914473309487248410248, 5.52539125504248435033352945178, 7.31264211607714213029084622909, 9.095459443738504631941897984034, 10.13059876841021225181537213266, 11.35403025669088024245444902074, 11.61752285197356155375713036741, 12.79854178371154154751381949555, 14.62975816372993029117991676272

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