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Properties

Label 1-85-85.24-r1-0-0
Degree $1$
Conductor $85$
Sign $-0.813 + 0.581i$
Analytic cond. $9.13451$
Root an. cond. $9.13451$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 0.707i)2-s + (0.382 + 0.923i)3-s + i·4-s + (−0.382 + 0.923i)6-s + (0.923 + 0.382i)7-s + (−0.707 + 0.707i)8-s + (−0.707 + 0.707i)9-s + (0.382 − 0.923i)11-s + (−0.923 + 0.382i)12-s + i·13-s + (0.382 + 0.923i)14-s − 16-s − 18-s + (−0.707 − 0.707i)19-s + i·21-s + (0.923 − 0.382i)22-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + (0.382 + 0.923i)3-s + i·4-s + (−0.382 + 0.923i)6-s + (0.923 + 0.382i)7-s + (−0.707 + 0.707i)8-s + (−0.707 + 0.707i)9-s + (0.382 − 0.923i)11-s + (−0.923 + 0.382i)12-s + i·13-s + (0.382 + 0.923i)14-s − 16-s − 18-s + (−0.707 − 0.707i)19-s + i·21-s + (0.923 − 0.382i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $-0.813 + 0.581i$
Analytic conductor: \(9.13451\)
Root analytic conductor: \(9.13451\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 85,\ (1:\ ),\ -0.813 + 0.581i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8458585200 + 2.636701922i\)
\(L(\frac12)\) \(\approx\) \(0.8458585200 + 2.636701922i\)
\(L(1)\) \(\approx\) \(1.222907019 + 1.329797016i\)
\(L(1)\) \(\approx\) \(1.222907019 + 1.329797016i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 + (0.382 + 0.923i)T \)
7 \( 1 + (0.923 + 0.382i)T \)
11 \( 1 + (0.382 - 0.923i)T \)
13 \( 1 + iT \)
19 \( 1 + (-0.707 - 0.707i)T \)
23 \( 1 + (0.382 - 0.923i)T \)
29 \( 1 + (0.923 - 0.382i)T \)
31 \( 1 + (0.382 + 0.923i)T \)
37 \( 1 + (0.382 + 0.923i)T \)
41 \( 1 + (-0.923 - 0.382i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
47 \( 1 + iT \)
53 \( 1 + (0.707 + 0.707i)T \)
59 \( 1 + (0.707 - 0.707i)T \)
61 \( 1 + (0.923 + 0.382i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.382 - 0.923i)T \)
73 \( 1 + (0.923 - 0.382i)T \)
79 \( 1 + (0.382 - 0.923i)T \)
83 \( 1 + (-0.707 - 0.707i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.923 + 0.382i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−30.13586291538842794892539013401, −29.49359577803000134022033645502, −28.13105688134720187889145980003, −27.20204209505094747087811499277, −25.422125393934195118527596556462, −24.62157050376255418914359441385, −23.48176001334890680968250617360, −22.841292922407903876603111457066, −21.248068953436316514788339921021, −20.30576023351376405946492529539, −19.56084422572777293208064890869, −18.25258635735802592412000074362, −17.38347807919265733124080167183, −15.14120479264674469119780416893, −14.42801022126311477536727140041, −13.280360910319779752789449224524, −12.345452993171040257348474892414, −11.28178553358783174371882528852, −9.91199781708802367215867869735, −8.26735675193185876685442882342, −6.94360471319390820094845839243, −5.44988301727569002248497975148, −3.92660736935808660828905231124, −2.3155096388601637717665413763, −1.097666725800881156866635167080, 2.623723210352441436055995810156, 4.15083963487392808727723600438, 5.06931258311018511964752096431, 6.52154182895959504493356685333, 8.299978202815468157583276487625, 8.94519356569426358490929517547, 10.90948614892395900405500010462, 11.93183215971682955283218943237, 13.71839713925782094831075561698, 14.44218737640870354698460080078, 15.4177344656181425648608459947, 16.47264075571347588190011853712, 17.40770679834893736125005951426, 19.07605639923805084328362238027, 20.66001000251621456266737823738, 21.48683995432357899273698262688, 22.10155760150345028650620336957, 23.571811578818818661830651326327, 24.535499292441368222185851331931, 25.542377493003169738142178093023, 26.68355852917175758568718516130, 27.31386634679520174093114500130, 28.74272319736784949352742046221, 30.401737231785802927633855354113, 31.08845370926954442904328683479

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