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Properties

Label 1-91-91.23-r0-0-0
Degree $1$
Conductor $91$
Sign $0.998 + 0.0505i$
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  − 2-s + (−0.5 − 0.866i)3-s + 4-s + (0.5 + 0.866i)5-s + (0.5 + 0.866i)6-s − 8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + (−0.5 − 0.866i)12-s + (0.5 − 0.866i)15-s + 16-s + 17-s + (0.5 − 0.866i)18-s + (0.5 − 0.866i)19-s + (0.5 + 0.866i)20-s + ⋯
L(s)  = 1  − 2-s + (−0.5 − 0.866i)3-s + 4-s + (0.5 + 0.866i)5-s + (0.5 + 0.866i)6-s − 8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + (−0.5 − 0.866i)12-s + (0.5 − 0.866i)15-s + 16-s + 17-s + (0.5 − 0.866i)18-s + (0.5 − 0.866i)19-s + (0.5 + 0.866i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0505i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0505i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.998 + 0.0505i$
Analytic conductor: \(0.422602\)
Root analytic conductor: \(0.422602\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 91,\ (0:\ ),\ 0.998 + 0.0505i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6189525352 + 0.01564782979i\)
\(L(\frac12)\) \(\approx\) \(0.6189525352 + 0.01564782979i\)
\(L(1)\) \(\approx\) \(0.6737431044 + 0.02563813716i\)
\(L(1)\) \(\approx\) \(0.6737431044 + 0.02563813716i\)

Euler product

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

−29.85323002567383631576997280886, −29.05312716107167798792425331159, −28.19220783893284317011916851830, −27.35804682955135807428354879639, −26.51984011708991059709083921281, −25.24587394184596959065413098752, −24.40896126626759839336763174899, −23.016861806223747792497901912020, −21.41210039308656788529416328969, −20.96401369784454803616003964300, −19.76879481922413436944190222392, −18.44741198626218765476656588043, −17.0910686809346011079428914580, −16.687950329753032488264289685584, −15.67878697343599619762470343586, −14.25459098006907109290787066474, −12.368855801455633809603626091022, −11.349835069579159907490023253332, −10.09043940791595797728547583762, −9.26904604152560764600081284021, −8.202254804272799999876358128025, −6.294582129999782837111333948381, −5.26693173960445399753025255898, −3.383884957983198444472620902645, −1.15667827207832141004561070345, 1.486947483904770784454049902447, 2.82656350221672046335978603427, 5.60045337146596548099485468493, 6.86740190747657730175177497178, 7.46943174329606446925094930231, 9.17457783517757716000414637177, 10.41148947174921093958429625651, 11.42425322230024525440796863645, 12.527951540519283986138528578871, 14.07958749217316743035447633078, 15.31367526563497872916031929585, 16.92999729487773414302919690965, 17.574085708964570297686315323946, 18.55359505690382256450042221041, 19.28269198137389846547375389416, 20.57430495014767577487458125612, 22.032733933025571055247748075677, 23.09786839583052909770749107479, 24.37576336192416242370675387357, 25.37883539044196171623868206684, 26.0302557236772986805756370489, 27.44058158380253938867262126010, 28.36572346105139507560420348153, 29.37744415262030012328627639321, 30.10592723016803878967743996993

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