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Properties

Label 2-85-85.84-c1-0-2
Degree $2$
Conductor $85$
Sign $-0.379 - 0.925i$
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  + 2.57i·2-s + 2.37·3-s − 4.64·4-s + (−1.95 − 1.08i)5-s + 6.12i·6-s + 1.53·7-s − 6.82i·8-s + 2.64·9-s + (2.79 − 5.04i)10-s − 3.95i·11-s − 11.0·12-s + 4.24i·13-s + 3.95i·14-s + (−4.64 − 2.57i)15-s + 8.29·16-s + (−3.21 − 2.57i)17-s + ⋯
L(s)  = 1  + 1.82i·2-s + 1.37·3-s − 2.32·4-s + (−0.874 − 0.485i)5-s + 2.50i·6-s + 0.579·7-s − 2.41i·8-s + 0.881·9-s + (0.884 − 1.59i)10-s − 1.19i·11-s − 3.18·12-s + 1.17i·13-s + 1.05i·14-s + (−1.19 − 0.665i)15-s + 2.07·16-s + (−0.780 − 0.625i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.379 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $-0.379 - 0.925i$
Analytic conductor: \(0.678728\)
Root analytic conductor: \(0.823849\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (84, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 85,\ (\ :1/2),\ -0.379 - 0.925i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.634464 + 0.945503i\)
\(L(\frac12)\) \(\approx\) \(0.634464 + 0.945503i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.95 + 1.08i)T \)
17 \( 1 + (3.21 + 2.57i)T \)
good2 \( 1 - 2.57iT - 2T^{2} \)
3 \( 1 - 2.37T + 3T^{2} \)
7 \( 1 - 1.53T + 7T^{2} \)
11 \( 1 + 3.95iT - 11T^{2} \)
13 \( 1 - 4.24iT - 13T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + 0.692T + 23T^{2} \)
29 \( 1 - 4.33iT - 29T^{2} \)
31 \( 1 - 1.78iT - 31T^{2} \)
37 \( 1 + 3.06T + 37T^{2} \)
41 \( 1 + 2.16iT - 41T^{2} \)
43 \( 1 + 4.24iT - 43T^{2} \)
47 \( 1 - 6.06iT - 47T^{2} \)
53 \( 1 + 5.15iT - 53T^{2} \)
59 \( 1 - 6T + 59T^{2} \)
61 \( 1 + 10.0iT - 61T^{2} \)
67 \( 1 - 4.24iT - 67T^{2} \)
71 \( 1 - 16.2iT - 71T^{2} \)
73 \( 1 - 8.66T + 73T^{2} \)
79 \( 1 + 1.78iT - 79T^{2} \)
83 \( 1 + 0.913iT - 83T^{2} \)
89 \( 1 - 4.93T + 89T^{2} \)
97 \( 1 - 11.1T + 97T^{2} \)
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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

−14.48162940512177847276907176916, −14.10893658439922283577666474424, −13.16754113512634457994550190849, −11.46455011086872498069795365583, −9.189790641716475196893380937277, −8.666188862532127306995089434564, −7.895799120571995791302387879684, −6.86018552501915105505399030637, −5.05152282648177571533560451126, −3.77999412062202688487984034550, 2.20995064530167775550758063233, 3.40764566604337659687836213943, 4.51755913816531217305794115046, 7.68373300389451020797009461455, 8.513038257217313726281397790273, 9.730902690072061875399874804286, 10.66881779841706634799649368121, 11.74269134574097268269674451698, 12.74068404755526734681786618546, 13.66092930278243816096946520329

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