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Properties

Label 2-85-5.4-c1-0-2
Degree $2$
Conductor $85$
Sign $-0.649 - 0.760i$
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  + 2.03i·2-s + 2.37i·3-s − 2.12·4-s + (1.70 − 1.45i)5-s − 4.81·6-s − 5.03i·7-s − 0.248i·8-s − 2.61·9-s + (2.94 + 3.45i)10-s − 1.90·11-s − 5.03i·12-s + 1.04i·13-s + 10.2·14-s + (3.44 + 4.03i)15-s − 3.74·16-s + i·17-s + ⋯
L(s)  = 1  + 1.43i·2-s + 1.36i·3-s − 1.06·4-s + (0.760 − 0.649i)5-s − 1.96·6-s − 1.90i·7-s − 0.0877i·8-s − 0.872·9-s + (0.932 + 1.09i)10-s − 0.575·11-s − 1.45i·12-s + 0.288i·13-s + 2.72·14-s + (0.888 + 1.04i)15-s − 0.935·16-s + 0.242i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.649 - 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.649 - 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $-0.649 - 0.760i$
Analytic conductor: \(0.678728\)
Root analytic conductor: \(0.823849\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 85,\ (\ :1/2),\ -0.649 - 0.760i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.427731 + 0.927818i\)
\(L(\frac12)\) \(\approx\) \(0.427731 + 0.927818i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.70 + 1.45i)T \)
17 \( 1 - iT \)
good2 \( 1 - 2.03iT - 2T^{2} \)
3 \( 1 - 2.37iT - 3T^{2} \)
7 \( 1 + 5.03iT - 7T^{2} \)
11 \( 1 + 1.90T + 11T^{2} \)
13 \( 1 - 1.04iT - 13T^{2} \)
19 \( 1 + 3.31T + 19T^{2} \)
23 \( 1 - 0.125iT - 23T^{2} \)
29 \( 1 - 5.56T + 29T^{2} \)
31 \( 1 + 4.99T + 31T^{2} \)
37 \( 1 - 1.56iT - 37T^{2} \)
41 \( 1 - 4.72T + 41T^{2} \)
43 \( 1 - 4.46iT - 43T^{2} \)
47 \( 1 + 1.04iT - 47T^{2} \)
53 \( 1 - 6.48iT - 53T^{2} \)
59 \( 1 - 2T + 59T^{2} \)
61 \( 1 - 7.14T + 61T^{2} \)
67 \( 1 + 3.28iT - 67T^{2} \)
71 \( 1 + 5.65T + 71T^{2} \)
73 \( 1 + 12.5iT - 73T^{2} \)
79 \( 1 - 13.3T + 79T^{2} \)
83 \( 1 + 1.14iT - 83T^{2} \)
89 \( 1 + 1.64T + 89T^{2} \)
97 \( 1 + 7.29iT - 97T^{2} \)
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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

−14.75500532772860077467201409098, −13.96828206438279548756549566498, −13.08505071885866961989442794343, −10.86049924214852962157987015529, −10.12016401943315962131028561430, −9.016461371775822346927275707281, −7.77221010300334358675990339190, −6.44375385686347397013938838855, −5.01790370209473958983877365535, −4.19908745218452527558406723933, 2.04510466587025927639381628107, 2.73871140181279190364178298869, 5.59578992718384232273264535218, 6.77106616517837562992908851303, 8.455805277852465827836392482348, 9.625804002588833263422329425616, 10.87070157758369689717090355683, 11.92382734734808769840283541731, 12.65638665661848521004038469692, 13.27727636331738859713042527528

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