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Properties

Label 2-85-17.16-c1-0-5
Degree $2$
Conductor $85$
Sign $-0.0519 + 0.998i$
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  − 0.311·2-s − 2.21i·3-s − 1.90·4-s − i·5-s + 0.688i·6-s − 1.59i·7-s + 1.21·8-s − 1.90·9-s + 0.311i·10-s + 1.31i·11-s + 4.21i·12-s + 3.52·13-s + 0.495i·14-s − 2.21·15-s + 3.42·16-s + (4.11 + 0.214i)17-s + ⋯
L(s)  = 1  − 0.219·2-s − 1.27i·3-s − 0.951·4-s − 0.447i·5-s + 0.281i·6-s − 0.601i·7-s + 0.429·8-s − 0.634·9-s + 0.0983i·10-s + 0.395i·11-s + 1.21i·12-s + 0.977·13-s + 0.132i·14-s − 0.571·15-s + 0.857·16-s + (0.998 + 0.0519i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0519 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0519 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $-0.0519 + 0.998i$
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: \((2,\ 85,\ (\ :1/2),\ -0.0519 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.514338 - 0.541806i\)
\(L(\frac12)\) \(\approx\) \(0.514338 - 0.541806i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.11 - 0.214i)T \)
good2 \( 1 + 0.311T + 2T^{2} \)
3 \( 1 + 2.21iT - 3T^{2} \)
7 \( 1 + 1.59iT - 7T^{2} \)
11 \( 1 - 1.31iT - 11T^{2} \)
13 \( 1 - 3.52T + 13T^{2} \)
19 \( 1 + 4.42T + 19T^{2} \)
23 \( 1 - 4.96iT - 23T^{2} \)
29 \( 1 - 8.42iT - 29T^{2} \)
31 \( 1 + 7.73iT - 31T^{2} \)
37 \( 1 + 7.05iT - 37T^{2} \)
41 \( 1 - 3.67iT - 41T^{2} \)
43 \( 1 - 2.47T + 43T^{2} \)
47 \( 1 - 3.33T + 47T^{2} \)
53 \( 1 + 9.18T + 53T^{2} \)
59 \( 1 - 1.37T + 59T^{2} \)
61 \( 1 - 15.4iT - 61T^{2} \)
67 \( 1 + 9.13T + 67T^{2} \)
71 \( 1 - 10.5iT - 71T^{2} \)
73 \( 1 + 5.57iT - 73T^{2} \)
79 \( 1 - 7.87iT - 79T^{2} \)
83 \( 1 - 7.19T + 83T^{2} \)
89 \( 1 - 11.6T + 89T^{2} \)
97 \( 1 - 15.4iT - 97T^{2} \)
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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

−13.61059560286705640288757263815, −13.05753456270580507069062245106, −12.21326687255411954437229094682, −10.66992904853510902013296802307, −9.340039156522349405263944628865, −8.182803325578237743418172480605, −7.30391525963402033418682859844, −5.75518219723049322944657147166, −4.04959341000931385143931604014, −1.25744698738074957478225882982, 3.48120839069573025536433432123, 4.70147651434325023226339488968, 6.05337320604467168464286677449, 8.216330727410611281711558704369, 9.069968567484042536616178931873, 10.13336588526159082620290851801, 10.86018071303374239414704672281, 12.36879340142317854561936825555, 13.70285252179947269782466285114, 14.63812795395812284185151452641

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