Properties

Degree 1
Conductor $ 7 \cdot 13 $
Sign $0.986 - 0.165i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(91\)    =    \(7 \cdot 13\)
\( \varepsilon \)  =  $0.986 - 0.165i$
motivic weight  =  \(0\)
character  :  $\chi_{91} (44, \cdot )$
Sato-Tate  :  $\mu(12)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 91,\ (1:\ ),\ 0.986 - 0.165i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.4502453063 - 0.03750679775i$
$L(\frac12,\chi)$  $\approx$  $0.4502453063 - 0.03750679775i$
$L(\chi,1)$  $\approx$  0.4517056539 + 0.1686716827i
$L(1,\chi)$  $\approx$  0.4517056539 + 0.1686716827i

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

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

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