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Properties

Label 2-85-85.19-c1-0-5
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

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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: $\chi_{85} (19, \cdot )$
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 \( 1 + (-0.210 + 0.0872i)T + (68.5 - 68.5i)T^{2} \)
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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

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

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