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Properties

Label 1-91-91.60-r1-0-0
Degree $1$
Conductor $91$
Sign $0.986 + 0.165i$
Analytic cond. $9.77930$
Root an. cond. $9.77930$
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  + (−0.866 − 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.5 + 0.866i)4-s + (−0.866 − 0.5i)5-s + i·6-s − i·8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)10-s + (−0.866 + 0.5i)11-s + (0.5 − 0.866i)12-s + i·15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (0.866 − 0.5i)18-s + (−0.866 − 0.5i)19-s − i·20-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.5 + 0.866i)4-s + (−0.866 − 0.5i)5-s + i·6-s − i·8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)10-s + (−0.866 + 0.5i)11-s + (0.5 − 0.866i)12-s + i·15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (0.866 − 0.5i)18-s + (−0.866 − 0.5i)19-s − i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.986 + 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.986 + 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.986 + 0.165i$
Analytic conductor: \(9.77930\)
Root analytic conductor: \(9.77930\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (60, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 91,\ (1:\ ),\ 0.986 + 0.165i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4502453063 + 0.03750679775i\)
\(L(\frac12)\) \(\approx\) \(0.4502453063 + 0.03750679775i\)
\(L(1)\) \(\approx\) \(0.4517056539 - 0.1686716827i\)
\(L(1)\) \(\approx\) \(0.4517056539 - 0.1686716827i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + T \)
31 \( 1 + (0.866 - 0.5i)T \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 + iT \)
43 \( 1 - T \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (-0.866 + 0.5i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + iT \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + iT \)
89 \( 1 + (0.866 + 0.5i)T \)
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

−29.74741345450309717505015778451, −28.79448180332516499261773967054, −27.68598876049446519730559501266, −27.02981073526746867739751479411, −26.31022926969690953358200395779, −25.12424011973936901623033443808, −23.41962227958723445816361760674, −23.242095674453105839284702247341, −21.56099879903396250967700782679, −20.42954188341572921783794469697, −19.1854869469401521746868783498, −18.25207574106029312148384800882, −17.020613445856502356989150525446, −15.95468801586658038088654444543, −15.39522759340045985961355807687, −14.20744565670176793026880047800, −11.94856182947687653804359271761, −10.919902657312548042504020098436, −10.13163507904890167989311874376, −8.720810081053008420539720940331, −7.53959980452166419034682575198, −6.16230375948430918031392623210, −4.8435241402637995449390850603, −3.0694994315606393054805922459, −0.37504682139744692063607602930, 0.99912308606199436046763169549, 2.63291718777986636693096824248, 4.58656127462464463645313055852, 6.53790158520010104521769810517, 7.80185446056302076536579458817, 8.48584366098387952644284748914, 10.301264741091131987987912818679, 11.366484727513495198418068288641, 12.43156044269101528515086233419, 13.06705804996957991973267497652, 15.250218272729924006517795607289, 16.50905203175466590565424422216, 17.34641133963784744841134015720, 18.52360324681232037020871893590, 19.3076550612888346045293164112, 20.2450988338665058531961291657, 21.46070021717478614056244481915, 22.99477051500649586680250254611, 23.85603405344561205626533524329, 24.991524996564017470881954588782, 26.06251106647043458239504417652, 27.32023466995039852172382321126, 28.35675026106238296420769662291, 28.76141592960638995675537180871, 30.205908715922857199025586091372

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