Properties

Label 1-91-91.41-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: $\chi_{91} (41, \cdot )$
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

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

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