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Properties

Label 1-91-91.18-r1-0-0
Degree $1$
Conductor $91$
Sign $0.728 + 0.684i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.728 + 0.684i$
Analytic conductor: \(9.77930\)
Root analytic conductor: \(9.77930\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 91,\ (1:\ ),\ 0.728 + 0.684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.614849474 + 1.035659125i\)
\(L(\frac12)\) \(\approx\) \(2.614849474 + 1.035659125i\)
\(L(1)\) \(\approx\) \(1.751839718 + 0.4113387713i\)
\(L(1)\) \(\approx\) \(1.751839718 + 0.4113387713i\)

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

−29.8303547769367453990593908150, −29.0469822174209261086585159984, −28.15508108855561127542434086473, −27.28890565566637860778400900304, −25.61213907427043054629641784599, −24.660866722218069523811969904819, −23.36947576318136044509262684117, −22.37918449425610295998153688160, −21.61844751776850909027400445256, −20.69226697430050716408920009854, −19.87292481757624562276060826228, −18.07248587681699622807267933456, −16.88774659789588489759338221948, −15.817202036163883584410731671818, −14.61166403563050044867242974938, −13.60579584300881016494404491119, −12.23569673448689109806984448560, −11.33693806100173321330549310668, −9.9124368048430713667409145597, −9.31971212702095408478168020625, −6.709065333580204594052797679658, −5.45115154909626713222807608181, −4.64378172183826716240834747817, −3.140294319917698542125760862020, −1.23452940176342957758022581948, 1.71008699301732976739282661570, 3.29787466036230658081505613353, 5.27999506743889555971022209047, 6.2379959482315684775056417625, 7.07517932619632986271185809181, 8.54162933220537183350946953898, 10.55873312363277819677685874429, 11.79679845850160082239455964967, 12.81774605109367203522031978511, 13.93066700211380984149139204721, 14.58124489210666405484845888769, 16.38398120893321858943283484233, 17.20936109909102026230907264954, 18.20950249957706883179354627775, 19.520084911669578087074244016601, 21.092770861302484655033700334944, 22.14223685581555231755170824025, 22.81280614071290520370257912285, 24.01459642463311021001696821660, 24.895236359966842340725332778081, 25.5955642370374034965558032154, 26.89951103121706809067520043499, 28.64977100863109301498541901064, 29.545546502660199355429771628732, 30.2726897329830076627029807510

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