Properties

Degree 1
Conductor $ 5 \cdot 17 $
Sign $0.307 - 0.951i$
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 + (0.923 − 0.382i)3-s i·4-s + (0.382 − 0.923i)6-s + (−0.382 + 0.923i)7-s + (−0.707 − 0.707i)8-s + (0.707 − 0.707i)9-s + (−0.382 + 0.923i)11-s + (−0.382 − 0.923i)12-s − 13-s + (0.382 + 0.923i)14-s − 16-s i·18-s + (0.707 + 0.707i)19-s + i·21-s + (0.382 + 0.923i)22-s + ⋯
L(s,χ)  = 1  + (0.707 − 0.707i)2-s + (0.923 − 0.382i)3-s i·4-s + (0.382 − 0.923i)6-s + (−0.382 + 0.923i)7-s + (−0.707 − 0.707i)8-s + (0.707 − 0.707i)9-s + (−0.382 + 0.923i)11-s + (−0.382 − 0.923i)12-s − 13-s + (0.382 + 0.923i)14-s − 16-s i·18-s + (0.707 + 0.707i)19-s + i·21-s + (0.382 + 0.923i)22-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.307 - 0.951i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.307 - 0.951i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(85\)    =    \(5 \cdot 17\)
\( \varepsilon \)  =  $0.307 - 0.951i$
motivic weight  =  \(0\)
character  :  $\chi_{85} (27, \cdot )$
Sato-Tate  :  $\mu(16)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 85,\ (0:\ ),\ 0.307 - 0.951i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.327798810 - 0.9658003181i$
$L(\frac12,\chi)$  $\approx$  $1.327798810 - 0.9658003181i$
$L(\chi,1)$  $\approx$  1.473158485 - 0.7501629931i
$L(1,\chi)$  $\approx$  1.473158485 - 0.7501629931i

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

−31.24802822999984717299614475040, −30.18330794128829640588648942663, −29.25392568171396179613232222983, −27.18942611963799027615201922781, −26.52817681883325226196715600071, −25.73481684413007716159391044346, −24.5161254404603392970113679441, −23.74606346614558312702162685246, −22.30419505187416065605404138289, −21.50412095562867417232849334368, −20.325340748723813793213427395609, −19.37355313668420440174909751415, −17.63805923476918361949367472939, −16.322853672151791080966913082193, −15.651471447783206449061243506736, −14.17231109634871721848329881767, −13.74998641991183462635052684025, −12.45731983551322646238575605451, −10.67256547901431183881246679781, −9.239634510145545230399927316, −7.9171163242466080508372545857, −6.98171776166756563953674282668, −5.202421944808067173500022900063, −3.892162293352105031949060949443, −2.806139491209099408946981551979, 2.00223220775164543782063320661, 2.94158764470949418054343373558, 4.50991886356931987137939068419, 6.07598696531026813814081308405, 7.622391639129300560229384519741, 9.29040317057024108173504705095, 10.09004028335683304506272399266, 12.10692010749123264592576960668, 12.55808368191689928885364278718, 13.90243411807633292287147558509, 14.87843971441156617667570364194, 15.748657964298213798429718393, 18.0064291514727372942732855273, 18.91810837060716155704360129418, 19.88369838634679223229504453496, 20.76551918691487595617075869444, 21.88060958239731966044464781625, 22.90228742486148594750288242675, 24.27606509199318988378252793277, 24.98017756106453420783880962062, 26.184004854707881457407834671097, 27.58476106773217820646817251153, 28.74542168315631984754279524086, 29.61678550889126669253307983460, 30.79767243830545073051625942238

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