Properties

Degree $1$
Conductor $91$
Sign $0.999 - 0.0375i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + i·2-s + (−0.5 + 0.866i)3-s − 4-s + (−0.866 − 0.5i)5-s + (−0.866 − 0.5i)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 + (0.866 − 0.5i)15-s + 16-s − 17-s + (0.866 − 0.5i)18-s + (0.866 − 0.5i)19-s + (0.866 + 0.5i)20-s + ⋯
L(s,χ)  = 1  + i·2-s + (−0.5 + 0.866i)3-s − 4-s + (−0.866 − 0.5i)5-s + (−0.866 − 0.5i)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 + (0.866 − 0.5i)15-s + 16-s − 17-s + (0.866 − 0.5i)18-s + (0.866 − 0.5i)19-s + (0.866 + 0.5i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.999 - 0.0375i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.999 - 0.0375i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.999 - 0.0375i$
Motivic weight: \(0\)
Character: $\chi_{91} (46, \cdot )$
Sato-Tate group: $\mu(12)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 91,\ (1:\ ),\ 0.999 - 0.0375i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.6356265095 + 0.01194645729i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.6356265095 + 0.01194645729i\)
\(L(\chi,1)\) \(\approx\) \(0.5823855984 + 0.3076617941i\)
\(L(1,\chi)\) \(\approx\) \(0.5823855984 + 0.3076617941i\)

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−30.11631440110346509418331112471, −29.29812849659782908447807133159, −28.20575006767634642763830669946, −27.30917411967729900919878301593, −26.29337233602426408537612002972, −24.58544011413190460732899556165, −23.623468105001233593304404399471, −22.55152135992932100010565547115, −22.013827128378420384386335223844, −20.173360507312202240452332335113, −19.459061153816189561985548119002, −18.533984855641599624457872521147, −17.66254423801616806717654411804, −16.27753909474791623988712570096, −14.48575420907922143436591159640, −13.49401765758018540819670923011, −12.12197343515599115733884613023, −11.560483784994570288345226309010, −10.51979942041702211398837199289, −8.77498357214516819284728767921, −7.5589975732424914055226760005, −6.095956927905613990310074469076, −4.35679243114058050134427680424, −2.91616828886730908262119832622, −1.224380782863076229195410390199, 0.36850103375058453678410952099, 3.896465100090453434409387593658, 4.62468018496037964137857283481, 5.998982900585529780379965115389, 7.37193079057648534394438006155, 8.79653683502309624879354980881, 9.67559707339583253429008005219, 11.35217199532715827395382786246, 12.46393510037638486996708551072, 14.0820442043343591148287113063, 15.368406867743075861953090186294, 15.87101467425224015785443480172, 16.99958554648530645137595190475, 17.81377593408914904337083661936, 19.459446109238196348971463134436, 20.59479548634250832078318185716, 22.16834070926495670839025572994, 22.70353508417886555331705208838, 23.87155244281167796661833392914, 24.7096373003145296566382468606, 26.19866010796777486811150698692, 26.93102778382724864215147912293, 27.91102929442649137793500921257, 28.49734977871951720587364110484, 30.46565195753053927729655771657

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