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Properties

Label 1-85-85.13-r1-0-0
Degree $1$
Conductor $85$
Sign $0.937 + 0.346i$
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  + i·2-s − 3-s − 4-s − i·6-s − 7-s − i·8-s + 9-s − i·11-s + 12-s + i·13-s − i·14-s + 16-s + i·18-s + 19-s + 21-s + 22-s + ⋯
L(s)  = 1  + i·2-s − 3-s − 4-s − i·6-s − 7-s − i·8-s + 9-s − i·11-s + 12-s + i·13-s − i·14-s + 16-s + i·18-s + 19-s + 21-s + 22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.937 + 0.346i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.937 + 0.346i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $0.937 + 0.346i$
Analytic conductor: \(9.13451\)
Root analytic conductor: \(9.13451\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 85,\ (1:\ ),\ 0.937 + 0.346i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7963675153 + 0.1425764269i\)
\(L(\frac12)\) \(\approx\) \(0.7963675153 + 0.1425764269i\)
\(L(1)\) \(\approx\) \(0.6391753272 + 0.2364463424i\)
\(L(1)\) \(\approx\) \(0.6391753272 + 0.2364463424i\)

Euler product

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

−30.13538883770969617709934887387, −29.269050802285173784442897854085, −28.485393306760800464355150782882, −27.65702905774258910708299443236, −26.561486532181760982950162729356, −25.16293908611639546331601715729, −23.508280799268630209801769515701, −22.66531792875190080275760587784, −22.12870850236098256868397025943, −20.71356534322019568565612597330, −19.699434786392007955292523096167, −18.442505462299100978609335585201, −17.63529073692694553825954702351, −16.435545866754361303825477462896, −15.0413282547146071127906240181, −13.147645689204958998031516544, −12.59062789094605295264438059302, −11.39860606697512778506208544054, −10.22151383202898222527037985906, −9.442057034944181529493737578802, −7.42022624838420150629061108149, −5.80274512418354134845429166226, −4.55448983503179872114500335361, −2.98904740628861449046046794740, −0.99603226929581287433480454052, 0.59796348203048444418493465645, 3.70759208840698001499535554834, 5.22951985121487537726807913261, 6.29980927200661733361577684695, 7.18814169763145963607359915347, 8.944677738985142604298719607118, 10.07016348811410285795949561901, 11.61014690472457162253572803372, 12.944374816816608833417818373009, 13.94087987442854171758747493959, 15.61338160617882056462930924504, 16.36140291563075261926750742113, 17.11878431700117113977929709049, 18.480396886846662752040622426011, 19.21485010637922696194092934457, 21.41421406119012037722502533006, 22.28042773645751813126214748850, 23.185970224283181277421354737079, 24.087303225673173468708862032261, 25.08293064031575955524977335154, 26.46731244447300942901660656981, 27.09238746536467187345576885291, 28.549571823074728401493767243926, 29.13989020687002340261679582992, 30.67361848124573131502858701971

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