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Properties

Label 2-85-17.16-c1-0-0
Degree $2$
Conductor $85$
Sign $0.615 - 0.787i$
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  − 2.17·2-s + 0.539i·3-s + 2.70·4-s − i·5-s − 1.17i·6-s + 4.87i·7-s − 1.53·8-s + 2.70·9-s + 2.17i·10-s + 3.17i·11-s + 1.46i·12-s + 2.63·13-s − 10.5i·14-s + 0.539·15-s − 2.07·16-s + (−3.24 − 2.53i)17-s + ⋯
L(s)  = 1  − 1.53·2-s + 0.311i·3-s + 1.35·4-s − 0.447i·5-s − 0.477i·6-s + 1.84i·7-s − 0.544·8-s + 0.903·9-s + 0.686i·10-s + 0.955i·11-s + 0.421i·12-s + 0.729·13-s − 2.82i·14-s + 0.139·15-s − 0.519·16-s + (−0.787 − 0.615i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 - 0.787i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.615 - 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $0.615 - 0.787i$
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: \((2,\ 85,\ (\ :1/2),\ 0.615 - 0.787i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.455653 + 0.222171i\)
\(L(\frac12)\) \(\approx\) \(0.455653 + 0.222171i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.24 + 2.53i)T \)
good2 \( 1 + 2.17T + 2T^{2} \)
3 \( 1 - 0.539iT - 3T^{2} \)
7 \( 1 - 4.87iT - 7T^{2} \)
11 \( 1 - 3.17iT - 11T^{2} \)
13 \( 1 - 2.63T + 13T^{2} \)
19 \( 1 - 1.07T + 19T^{2} \)
23 \( 1 + 5.21iT - 23T^{2} \)
29 \( 1 - 2.92iT - 29T^{2} \)
31 \( 1 + 4.09iT - 31T^{2} \)
37 \( 1 + 5.26iT - 37T^{2} \)
41 \( 1 - 5.60iT - 41T^{2} \)
43 \( 1 - 3.36T + 43T^{2} \)
47 \( 1 + 6.78T + 47T^{2} \)
53 \( 1 - 3.75T + 53T^{2} \)
59 \( 1 + 2.34T + 59T^{2} \)
61 \( 1 + 12.2iT - 61T^{2} \)
67 \( 1 - 10.2T + 67T^{2} \)
71 \( 1 + 4.06iT - 71T^{2} \)
73 \( 1 + 11.0iT - 73T^{2} \)
79 \( 1 + 6.92iT - 79T^{2} \)
83 \( 1 - 8.23T + 83T^{2} \)
89 \( 1 - 7.15T + 89T^{2} \)
97 \( 1 - 8.18iT - 97T^{2} \)
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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

−14.95296491144703727800965362269, −13.02572931974601969823710537263, −12.04275257309838249969100381306, −10.87138738782782313888309989145, −9.544123839418434955514328880299, −9.103096593238461609267116044419, −8.017770023677950638430227825144, −6.56523488289068496718447896444, −4.81491545421468211833334731675, −2.07961361918891517013062975765, 1.21563611534517583450234751434, 3.91769165052456558310298023946, 6.60113214829784771655336847090, 7.38525042743959833248473495426, 8.373305896050367527806367713962, 9.824598648838181641358882583407, 10.62113558302734283458921638363, 11.29900725424366086247172431906, 13.29713450455463627154021523586, 13.83206094318183029450694449026

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