↑

Properties

Label 2-85-85.9-c1-0-1
Degree $2$
Conductor $85$
Sign $0.0763 - 0.997i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.672 + 0.672i)2-s + (−0.947 + 2.28i)3-s − 1.09i·4-s + (−0.734 + 2.11i)5-s + (−2.17 + 0.901i)6-s + (0.492 − 0.204i)7-s + (2.08 − 2.08i)8-s + (−2.20 − 2.20i)9-s + (−1.91 + 0.926i)10-s + (4.31 − 1.78i)11-s + (2.50 + 1.03i)12-s − 3.92·13-s + (0.468 + 0.194i)14-s + (−4.13 − 3.68i)15-s + 0.614·16-s + (4.03 − 0.867i)17-s + ⋯
L(s)  = 1  + (0.475 + 0.475i)2-s + (−0.546 + 1.32i)3-s − 0.547i·4-s + (−0.328 + 0.944i)5-s + (−0.888 + 0.367i)6-s + (0.186 − 0.0771i)7-s + (0.736 − 0.736i)8-s + (−0.736 − 0.736i)9-s + (−0.605 + 0.293i)10-s + (1.30 − 0.539i)11-s + (0.722 + 0.299i)12-s − 1.08·13-s + (0.125 + 0.0519i)14-s + (−1.06 − 0.950i)15-s + 0.153·16-s + (0.977 − 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0763 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0763 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $0.0763 - 0.997i$
Analytic conductor: \(0.678728\)
Root analytic conductor: \(0.823849\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 85,\ (\ :1/2),\ 0.0763 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.763622 + 0.707402i\)
\(L(\frac12)\) \(\approx\) \(0.763622 + 0.707402i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.734 - 2.11i)T \)
17 \( 1 + (-4.03 + 0.867i)T \)
good2 \( 1 + (-0.672 - 0.672i)T + 2iT^{2} \)
3 \( 1 + (0.947 - 2.28i)T + (-2.12 - 2.12i)T^{2} \)
7 \( 1 + (-0.492 + 0.204i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (-4.31 + 1.78i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + 3.92T + 13T^{2} \)
19 \( 1 + (-0.708 + 0.708i)T - 19iT^{2} \)
23 \( 1 + (2.19 + 5.30i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (3.17 - 7.65i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (2.29 + 0.948i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (1.56 - 3.78i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (0.311 + 0.752i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (-4.55 + 4.55i)T - 43iT^{2} \)
47 \( 1 - 1.70T + 47T^{2} \)
53 \( 1 + (-5.77 - 5.77i)T + 53iT^{2} \)
59 \( 1 + (4.28 + 4.28i)T + 59iT^{2} \)
61 \( 1 + (0.432 + 1.04i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + 7.68iT - 67T^{2} \)
71 \( 1 + (-6.41 - 2.65i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (11.1 + 4.62i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (1.36 - 0.566i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (-11.4 - 11.4i)T + 83iT^{2} \)
89 \( 1 + 12.0iT - 89T^{2} \)
97 \( 1 + (-0.210 - 0.0872i)T + (68.5 + 68.5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.53041613901997051727797124819, −14.18719824965621424311921444810, −12.13472388255126356960118454072, −11.02009103416678294638026362701, −10.28523257835607238839065608871, −9.334106653105715230028621366422, −7.27381461395879417982879893324, −6.08541919679992414669116616928, −4.88296498382985701553068765881, −3.70468341197974706217935604574, 1.73048144185212838484214803637, 4.05647116033177056536610726557, 5.54748654436376226200775698404, 7.25572320675905657850738942497, 7.968577534912149349699248915413, 9.518688751026352578855095712826, 11.62898185312708870127445809191, 11.97191094099371355707207114294, 12.64374735958038732871290547067, 13.54218158213958413741609741063

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