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Properties

Label 1-85-85.44-r1-0-0
Degree $1$
Conductor $85$
Sign $-0.971 + 0.238i$
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

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

Invariants

Degree: \(1\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $-0.971 + 0.238i$
Analytic conductor: \(9.13451\)
Root analytic conductor: \(9.13451\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 85,\ (1:\ ),\ -0.971 + 0.238i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08428288821 + 0.6975239437i\)
\(L(\frac12)\) \(\approx\) \(0.08428288821 + 0.6975239437i\)
\(L(1)\) \(\approx\) \(0.8484621201 + 0.3415124201i\)
\(L(1)\) \(\approx\) \(0.8484621201 + 0.3415124201i\)

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

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

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