Properties

Degree 1
Conductor $ 5 \cdot 17 $
Sign $-0.813 + 0.581i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(85\)    =    \(5 \cdot 17\)
\( \varepsilon \)  =  $-0.813 + 0.581i$
motivic weight  =  \(0\)
character  :  $\chi_{85} (24, \cdot )$
Sato-Tate  :  $\mu(16)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 85,\ (1:\ ),\ -0.813 + 0.581i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.8458585200 + 2.636701922i$
$L(\frac12,\chi)$  $\approx$  $0.8458585200 + 2.636701922i$
$L(\chi,1)$  $\approx$  1.222907019 + 1.329797016i
$L(1,\chi)$  $\approx$  1.222907019 + 1.329797016i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

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