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Properties

Label 2-91-13.12-c1-0-2
Degree $2$
Conductor $91$
Sign $0.601 - 0.798i$
Analytic cond. $0.726638$
Root an. cond. $0.852431$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.17i·2-s + 0.539·3-s + 0.630·4-s − 0.460i·5-s + 0.630i·6-s − i·7-s + 3.07i·8-s − 2.70·9-s + 0.539·10-s + 0.829i·11-s + 0.340·12-s + (−2.87 − 2.17i)13-s + 1.17·14-s − 0.248i·15-s − 2.34·16-s + 2.87·17-s + ⋯
L(s)  = 1  + 0.827i·2-s + 0.311·3-s + 0.315·4-s − 0.206i·5-s + 0.257i·6-s − 0.377i·7-s + 1.08i·8-s − 0.903·9-s + 0.170·10-s + 0.250i·11-s + 0.0981·12-s + (−0.798 − 0.601i)13-s + 0.312·14-s − 0.0641i·15-s − 0.585·16-s + 0.698·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.601 - 0.798i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.601 - 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.601 - 0.798i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.601 - 0.798i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01036 + 0.503701i\)
\(L(\frac12)\) \(\approx\) \(1.01036 + 0.503701i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + iT \)
13 \( 1 + (2.87 + 2.17i)T \)
good2 \( 1 - 1.17iT - 2T^{2} \)
3 \( 1 - 0.539T + 3T^{2} \)
5 \( 1 + 0.460iT - 5T^{2} \)
11 \( 1 - 0.829iT - 11T^{2} \)
17 \( 1 - 2.87T + 17T^{2} \)
19 \( 1 + 4.32iT - 19T^{2} \)
23 \( 1 + 5.04T + 23T^{2} \)
29 \( 1 - 0.261T + 29T^{2} \)
31 \( 1 + 6.80iT - 31T^{2} \)
37 \( 1 - 9.51iT - 37T^{2} \)
41 \( 1 - 6.68iT - 41T^{2} \)
43 \( 1 - 0.418T + 43T^{2} \)
47 \( 1 - 9.24iT - 47T^{2} \)
53 \( 1 - 1.63T + 53T^{2} \)
59 \( 1 + 2.78iT - 59T^{2} \)
61 \( 1 - 7.26T + 61T^{2} \)
67 \( 1 - 5.07iT - 67T^{2} \)
71 \( 1 + 12.7iT - 71T^{2} \)
73 \( 1 - 0.353iT - 73T^{2} \)
79 \( 1 - 2.81T + 79T^{2} \)
83 \( 1 + 10.3iT - 83T^{2} \)
89 \( 1 - 5.43iT - 89T^{2} \)
97 \( 1 - 12.6iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.55489268704814568117068852782, −13.47806499756116549839170074580, −12.11445807734131386658865890524, −11.09276162873804858964759355944, −9.745783686367274712587899073540, −8.331302842817510185479288024470, −7.53539649918199473425592615145, −6.22676806309552780090629189307, −4.96348232972901228501914286517, −2.76963395708607092435885909491, 2.27275681099378815718370678345, 3.58782396749804202945136524624, 5.68127986712893890694266321547, 7.11616696485486909006267447342, 8.535958805173087860680457716634, 9.780408507737456583356864958378, 10.78490521447773331975272410428, 11.90485926727221512169185059426, 12.45523679374304928334475730042, 14.05068252754954068192883691884

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