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Properties

Label 1-85-85.2-r1-0-0
Degree $1$
Conductor $85$
Sign $0.485 - 0.874i$
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  − 2-s + (−0.707 − 0.707i)3-s + 4-s + (0.707 + 0.707i)6-s + (0.707 − 0.707i)7-s − 8-s + i·9-s + (0.707 + 0.707i)11-s + (−0.707 − 0.707i)12-s + i·13-s + (−0.707 + 0.707i)14-s + 16-s − i·18-s − i·19-s − 21-s + (−0.707 − 0.707i)22-s + ⋯
L(s)  = 1  − 2-s + (−0.707 − 0.707i)3-s + 4-s + (0.707 + 0.707i)6-s + (0.707 − 0.707i)7-s − 8-s + i·9-s + (0.707 + 0.707i)11-s + (−0.707 − 0.707i)12-s + i·13-s + (−0.707 + 0.707i)14-s + 16-s − i·18-s − i·19-s − 21-s + (−0.707 − 0.707i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.485 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.485 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $0.485 - 0.874i$
Analytic conductor: \(9.13451\)
Root analytic conductor: \(9.13451\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 85,\ (1:\ ),\ 0.485 - 0.874i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7833983283 - 0.4609829046i\)
\(L(\frac12)\) \(\approx\) \(0.7833983283 - 0.4609829046i\)
\(L(1)\) \(\approx\) \(0.6552752360 - 0.1872600781i\)
\(L(1)\) \(\approx\) \(0.6552752360 - 0.1872600781i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
17 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (-0.707 - 0.707i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
11 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + iT \)
19 \( 1 - iT \)
23 \( 1 + (0.707 - 0.707i)T \)
29 \( 1 + (0.707 - 0.707i)T \)
31 \( 1 + (0.707 - 0.707i)T \)
37 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 + (-0.707 - 0.707i)T \)
43 \( 1 - T \)
47 \( 1 - iT \)
53 \( 1 + T \)
59 \( 1 + iT \)
61 \( 1 + (-0.707 - 0.707i)T \)
67 \( 1 + iT \)
71 \( 1 + (0.707 - 0.707i)T \)
73 \( 1 + (0.707 + 0.707i)T \)
79 \( 1 + (-0.707 - 0.707i)T \)
83 \( 1 + T \)
89 \( 1 + T \)
97 \( 1 + (-0.707 - 0.707i)T \)
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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

−30.314031795164744739008107194995, −29.30179402333211565732270344877, −28.3421429976376531877305404041, −27.23403797606684176405914985635, −27.12255615421328640770464424429, −25.39373991314070522331572407692, −24.582080390254377036312065152416, −23.22841099127827577468152893903, −21.79111642061991787778286406045, −21.06546068743431416671799833025, −19.83307618955628609864527583326, −18.45285045593323549063307244188, −17.597092138614138189897198911185, −16.62827935005594759739419676076, −15.5647198294258681607023919429, −14.62150484261558793713302513992, −12.28767368057502739504808960030, −11.381565052584968141483091721392, −10.41413955352695852481313430498, −9.17588841084002884507208584087, −8.111924088683160970153428614145, −6.35856799748482159972090840437, −5.27725480453999825071742208815, −3.25191008899379840390193298697, −1.1383928669078461898728057337, 0.82207299266547469513904884958, 2.0927668191575769470539189485, 4.63520506642173997669643698013, 6.53258958451520037386644414331, 7.23186575617702492602606225595, 8.5538608238709414943197369519, 10.06755472260077860976640012735, 11.29150273280080257914189834735, 11.98260000413246335450481872326, 13.61021681615192295928604314233, 15.071306401302339897487255872790, 16.712875157987187688728003316291, 17.22436845361609205357524964232, 18.227073448434889826842660019920, 19.29346997590011269633793210145, 20.29450250841027670367286308510, 21.60900884543719912345772009101, 23.13260363322880235981397961304, 24.126078078360455828348576511477, 24.922292908204866899735347782387, 26.21090642838581081596960572018, 27.28983070683064921500005966461, 28.256246629327622456307699586154, 29.04609075099328075872996652390, 30.25677765074906886998491640916

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