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Properties

Label 1-91-91.20-r0-0-0
Degree $1$
Conductor $91$
Sign $0.878 - 0.477i$
Analytic cond. $0.422602$
Root an. cond. $0.422602$
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 − i·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 + 12-s + (0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + i·18-s + (0.866 + 0.5i)19-s + (−0.866 − 0.5i)20-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.5 + 0.866i)3-s + (0.5 − 0.866i)4-s − i·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 + 12-s + (0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + i·18-s + (0.866 + 0.5i)19-s + (−0.866 − 0.5i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.878 - 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.878 - 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.878 - 0.477i$
Analytic conductor: \(0.422602\)
Root analytic conductor: \(0.422602\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 91,\ (0:\ ),\ 0.878 - 0.477i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.652571179 - 0.4200425457i\)
\(L(\frac12)\) \(\approx\) \(1.652571179 - 0.4200425457i\)
\(L(1)\) \(\approx\) \(1.665845087 - 0.3069524867i\)
\(L(1)\) \(\approx\) \(1.665845087 - 0.3069524867i\)

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 \( 1 - iT \)
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 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 - iT \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + iT \)
53 \( 1 + T \)
59 \( 1 + (-0.866 - 0.5i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (-0.866 - 0.5i)T \)
73 \( 1 + iT \)
79 \( 1 + T \)
83 \( 1 - iT \)
89 \( 1 + (0.866 - 0.5i)T \)
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

−30.84107226917233277216181730097, −29.66698971938739183061533748966, −29.02921554034919847813594730705, −26.71101050853442179291902804394, −26.18193083971735843285833551923, −25.06899164397720869179572637039, −24.1840251259247023133502422857, −23.19104524506203626267824529445, −22.29661999836834171709895198332, −21.041326964849374951523150234000, −19.913240100826149382504601331073, −18.534432123252970520930523020010, −17.77586093302064198522636275797, −16.10609196135399605568354381082, −14.99968861099556897846674713113, −14.003668740117024362816894775272, −13.28819122585187507002922198109, −11.98726285619947736920124412345, −10.8333560672877567674106584024, −8.74704801779522948947409424614, −7.41197207815699498309169426653, −6.75286990515015679735266249277, −5.341187682779361946310846626345, −3.33812292534978699923431989675, −2.47942833141173403490941725991, 1.99204759908606176728482962656, 3.589659099768007550961312758038, 4.72171601585686785922124775945, 5.64814731279186132841439008257, 7.82888964372272470280143554175, 9.32476208272347411867077214622, 10.273734529796449773630932750790, 11.58411203279914788471364147856, 12.91993341275645628474758285569, 13.74360179780641005360957449202, 15.1873390545656902067849031003, 15.791201507350109230901744082597, 17.10349906174011921936290849179, 19.01023592267686498436912208522, 20.18856943258609644291439026565, 20.73354232518425146173896613813, 21.62474686725878991103073686804, 22.75643543273837302784432134636, 23.921560999210576989276842223230, 24.92246689764394764095058369516, 26.0607803769492521940208771455, 27.5005093446050686239538349808, 28.360830075736739827008607758699, 29.13168676288259457612255989871, 30.77049334996076189305876334041

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