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Properties

Label 2-85-17.16-c1-0-3
Degree $2$
Conductor $85$
Sign $0.891 - 0.453i$
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  + 1.48·2-s + 1.67i·3-s + 0.193·4-s − i·5-s + 2.48i·6-s − 1.28i·7-s − 2.67·8-s + 0.193·9-s − 1.48i·10-s − 0.481i·11-s + 0.324i·12-s − 2.15·13-s − 1.90i·14-s + 1.67·15-s − 4.35·16-s + (−1.86 − 3.67i)17-s + ⋯
L(s)  = 1  + 1.04·2-s + 0.967i·3-s + 0.0969·4-s − 0.447i·5-s + 1.01i·6-s − 0.486i·7-s − 0.945·8-s + 0.0646·9-s − 0.468i·10-s − 0.145i·11-s + 0.0937i·12-s − 0.598·13-s − 0.509i·14-s + 0.432·15-s − 1.08·16-s + (−0.453 − 0.891i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 - 0.453i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.891 - 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $0.891 - 0.453i$
Analytic conductor: \(0.678728\)
Root analytic conductor: \(0.823849\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 85,\ (\ :1/2),\ 0.891 - 0.453i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35062 + 0.323714i\)
\(L(\frac12)\) \(\approx\) \(1.35062 + 0.323714i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + iT \)
17 \( 1 + (1.86 + 3.67i)T \)
good2 \( 1 - 1.48T + 2T^{2} \)
3 \( 1 - 1.67iT - 3T^{2} \)
7 \( 1 + 1.28iT - 7T^{2} \)
11 \( 1 + 0.481iT - 11T^{2} \)
13 \( 1 + 2.15T + 13T^{2} \)
19 \( 1 - 3.35T + 19T^{2} \)
23 \( 1 - 8.24iT - 23T^{2} \)
29 \( 1 - 0.649iT - 29T^{2} \)
31 \( 1 - 1.83iT - 31T^{2} \)
37 \( 1 - 4.31iT - 37T^{2} \)
41 \( 1 + 11.2iT - 41T^{2} \)
43 \( 1 - 8.15T + 43T^{2} \)
47 \( 1 + 6.54T + 47T^{2} \)
53 \( 1 + 8.57T + 53T^{2} \)
59 \( 1 - 4.96T + 59T^{2} \)
61 \( 1 - 2.83iT - 61T^{2} \)
67 \( 1 - 4.93T + 67T^{2} \)
71 \( 1 + 14.5iT - 71T^{2} \)
73 \( 1 + 13.3iT - 73T^{2} \)
79 \( 1 - 9.05iT - 79T^{2} \)
83 \( 1 + 13.4T + 83T^{2} \)
89 \( 1 + 16.7T + 89T^{2} \)
97 \( 1 + 3.66iT - 97T^{2} \)
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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

−14.16517607489399483724815141013, −13.47895676742927443618392997738, −12.35138725268567894129059712001, −11.27367396077437763918568005587, −9.838224123566889744365435042512, −9.095226182285639893665926635579, −7.27771140061484289861231126271, −5.44513492141751701841600746462, −4.59821217723280761032342108846, −3.44635277789694562052469751827, 2.56351429255536338061609757318, 4.38633722353706628401005267331, 5.92519258887829335811162084485, 6.92008023842215113872017073559, 8.325741252784406187119341839684, 9.799649494110798942877489755070, 11.39843034421646626310923301858, 12.53754108055622633719613888194, 12.90114044787476353186980602519, 14.13046821153905519972569027031

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