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Properties

Label 2-85-17.8-c1-0-0
Degree $2$
Conductor $85$
Sign $-0.991 + 0.131i$
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  + (−1.44 + 1.44i)2-s + (−1.22 + 2.95i)3-s − 2.18i·4-s + (0.923 + 0.382i)5-s + (−2.50 − 6.03i)6-s + (1.08 − 0.450i)7-s + (0.263 + 0.263i)8-s + (−5.09 − 5.09i)9-s + (−1.88 + 0.782i)10-s + (1.88 + 4.54i)11-s + (6.44 + 2.66i)12-s − 2.46i·13-s + (−0.921 + 2.22i)14-s + (−2.25 + 2.25i)15-s + 3.60·16-s + (−1.36 + 3.89i)17-s + ⋯
L(s)  = 1  + (−1.02 + 1.02i)2-s + (−0.706 + 1.70i)3-s − 1.09i·4-s + (0.413 + 0.171i)5-s + (−1.02 − 2.46i)6-s + (0.411 − 0.170i)7-s + (0.0931 + 0.0931i)8-s + (−1.69 − 1.69i)9-s + (−0.597 + 0.247i)10-s + (0.567 + 1.37i)11-s + (1.85 + 0.770i)12-s − 0.683i·13-s + (−0.246 + 0.594i)14-s + (−0.583 + 0.583i)15-s + 0.900·16-s + (−0.330 + 0.943i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.131i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $-0.991 + 0.131i$
Analytic conductor: \(0.678728\)
Root analytic conductor: \(0.823849\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 85,\ (\ :1/2),\ -0.991 + 0.131i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0319985 - 0.484762i\)
\(L(\frac12)\) \(\approx\) \(0.0319985 - 0.484762i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.923 - 0.382i)T \)
17 \( 1 + (1.36 - 3.89i)T \)
good2 \( 1 + (1.44 - 1.44i)T - 2iT^{2} \)
3 \( 1 + (1.22 - 2.95i)T + (-2.12 - 2.12i)T^{2} \)
7 \( 1 + (-1.08 + 0.450i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (-1.88 - 4.54i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + 2.46iT - 13T^{2} \)
19 \( 1 + (1.44 - 1.44i)T - 19iT^{2} \)
23 \( 1 + (0.0455 + 0.109i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (-0.984 - 0.407i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (-1.06 + 2.58i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (-0.885 + 2.13i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (0.662 - 0.274i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (-7.13 - 7.13i)T + 43iT^{2} \)
47 \( 1 + 3.39iT - 47T^{2} \)
53 \( 1 + (-9.84 + 9.84i)T - 53iT^{2} \)
59 \( 1 + (-1.07 - 1.07i)T + 59iT^{2} \)
61 \( 1 + (-7.46 + 3.09i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 - 4.92T + 67T^{2} \)
71 \( 1 + (-2.53 + 6.13i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-3.38 - 1.40i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (-3.13 - 7.55i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (4.30 - 4.30i)T - 83iT^{2} \)
89 \( 1 + 8.46iT - 89T^{2} \)
97 \( 1 + (4.14 + 1.71i)T + (68.5 + 68.5i)T^{2} \)
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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

−15.00198598864775666224815424052, −14.68606188558900471603478833566, −12.46405458576038021828335615739, −10.99663981904043413097439351990, −10.06340715409907134188378050064, −9.489602763268070972408244148188, −8.267874760419052620853422110532, −6.64893425705664633619425601090, −5.51748704203888310153410823974, −4.12106162840519913050752844776, 0.987172650165626962243973530006, 2.40705621098695212849327379128, 5.62317662787804155182241265765, 6.83587164202635661792252859864, 8.265427891099044015908694856375, 9.086803990336293222952378772307, 10.82421139318622319252396349576, 11.54997161356360265660410282282, 12.17286195377468432756849144222, 13.42996918896611198775031012923

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