Properties

Label 1-91-91.2-r1-0-0
Degree $1$
Conductor $91$
Sign $0.999 + 0.0375i$
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  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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.999 + 0.0375i$
Analytic conductor: \(9.77930\)
Root analytic conductor: \(9.77930\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 91,\ (1:\ ),\ 0.999 + 0.0375i)\)

Particular Values

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

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 - iT \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (0.866 - 0.5i)T \)
17 \( 1 - T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (0.866 + 0.5i)T \)
37 \( 1 + iT \)
41 \( 1 + (0.866 + 0.5i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + iT \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (-0.866 + 0.5i)T \)
73 \( 1 + (-0.866 - 0.5i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 - iT \)
89 \( 1 + iT \)
97 \( 1 + (-0.866 + 0.5i)T \)
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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

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

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