Properties

Degree 1
Conductor $ 5 \cdot 17 $
Sign $-0.971 - 0.238i$
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

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.971 - 0.238i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.971 - 0.238i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(85\)    =    \(5 \cdot 17\)
\( \varepsilon \)  =  $-0.971 - 0.238i$
motivic weight  =  \(0\)
character  :  $\chi_{85} (29, \cdot )$
Sato-Tate  :  $\mu(16)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 85,\ (1:\ ),\ -0.971 - 0.238i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.08428288821 - 0.6975239437i$
$L(\frac12,\chi)$  $\approx$  $0.08428288821 - 0.6975239437i$
$L(\chi,1)$  $\approx$  0.8484621201 - 0.3415124201i
$L(1,\chi)$  $\approx$  0.8484621201 - 0.3415124201i

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

−31.082257522570888176521274898727, −30.0087542477778001274618959275, −29.19899451970301869646754693502, −27.98387062322860545957743604685, −26.07007108642166389096196747527, −25.73663050946511734811613815114, −24.359730306169920146842152964003, −23.51835170926091829468107818642, −22.798985261848515715478762239604, −21.72561101759600641309493827324, −20.2032682333376090039485404387, −18.97249146040002216898383447343, −17.65675456967455655235701638516, −16.78862358555063498330508334650, −15.6836293884975624492304797844, −14.248822691502328406520540999855, −13.16794046751363373072871805129, −12.49651068878330377955333965821, −11.19113197580899036878420197074, −9.24331921405626248349265703368, −7.54672250581950123498402620217, −6.83530708241126619966657062326, −5.67797727480432926481347265888, −4.12762894027290627530932383362, −2.323594842182056232070891167616, 0.25539352016611766957860884008, 2.82761371706617083244131832812, 3.8561034442410641005057425909, 5.41360665654368069276077476716, 6.21254665421140453491957610477, 8.736746072918089692452609718955, 10.062966527177450725666023386815, 10.78392132509770389300368741198, 12.12245116916854450681470219582, 13.13607111702238505845176981864, 14.58945428265573275749599837327, 15.61032586118535189656344256099, 16.5353096483401407405882222699, 18.26930387384576703220677981173, 19.431059748215017896288503701517, 20.58110515714471798701768962443, 21.503787010635692398843048415072, 22.44337046291438766128959608789, 23.08067960439896113325971349080, 24.45938226758349736287756022935, 25.86623861159095339179826340723, 27.16463811123021588679606362628, 28.07285720910140173991994302959, 29.04251693823860048351878980163, 29.76629297291680141940507673550

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