Properties

Label 2-91-91.83-c1-0-2
Degree $2$
Conductor $91$
Sign $0.927 - 0.373i$
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  + (−0.403 + 0.403i)2-s − 1.23i·3-s + 1.67i·4-s + (1.03 + 1.03i)5-s + (0.496 + 0.496i)6-s + (2.60 − 0.450i)7-s + (−1.48 − 1.48i)8-s + 1.48·9-s − 0.832·10-s + (−0.596 − 0.596i)11-s + 2.06·12-s + (−3.59 − 0.296i)13-s + (−0.869 + 1.23i)14-s + (1.27 − 1.27i)15-s − 2.15·16-s − 7.34·17-s + ⋯
L(s)  = 1  + (−0.284 + 0.284i)2-s − 0.711i·3-s + 0.837i·4-s + (0.461 + 0.461i)5-s + (0.202 + 0.202i)6-s + (0.985 − 0.170i)7-s + (−0.523 − 0.523i)8-s + 0.493·9-s − 0.263·10-s + (−0.179 − 0.179i)11-s + 0.595·12-s + (−0.996 − 0.0822i)13-s + (−0.232 + 0.329i)14-s + (0.328 − 0.328i)15-s − 0.539·16-s − 1.78·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.927 - 0.373i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.927 - 0.373i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.930231 + 0.180287i\)
\(L(\frac12)\) \(\approx\) \(0.930231 + 0.180287i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.60 + 0.450i)T \)
13 \( 1 + (3.59 + 0.296i)T \)
good2 \( 1 + (0.403 - 0.403i)T - 2iT^{2} \)
3 \( 1 + 1.23iT - 3T^{2} \)
5 \( 1 + (-1.03 - 1.03i)T + 5iT^{2} \)
11 \( 1 + (0.596 + 0.596i)T + 11iT^{2} \)
17 \( 1 + 7.34T + 17T^{2} \)
19 \( 1 + (-3.59 - 3.59i)T + 19iT^{2} \)
23 \( 1 + 4.44iT - 23T^{2} \)
29 \( 1 + 3.54T + 29T^{2} \)
31 \( 1 + (1.27 + 1.27i)T + 31iT^{2} \)
37 \( 1 + (-2.88 - 2.88i)T + 37iT^{2} \)
41 \( 1 + (-1.23 - 1.23i)T + 41iT^{2} \)
43 \( 1 - 8.66iT - 43T^{2} \)
47 \( 1 + (-2.52 + 2.52i)T - 47iT^{2} \)
53 \( 1 + 9.79T + 53T^{2} \)
59 \( 1 + (1.08 - 1.08i)T - 59iT^{2} \)
61 \( 1 - 7.10iT - 61T^{2} \)
67 \( 1 + (-8.76 + 8.76i)T - 67iT^{2} \)
71 \( 1 + (1.46 - 1.46i)T - 71iT^{2} \)
73 \( 1 + (0.103 - 0.103i)T - 73iT^{2} \)
79 \( 1 + 4.79T + 79T^{2} \)
83 \( 1 + (-12.3 - 12.3i)T + 83iT^{2} \)
89 \( 1 + (6.89 - 6.89i)T - 89iT^{2} \)
97 \( 1 + (6.05 + 6.05i)T + 97iT^{2} \)
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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

−14.04633240887764605162262994793, −13.08290252800128772165067280210, −12.17057264758700303978476538851, −11.04589238070926497643732897455, −9.673631824027728460457508963920, −8.251616470654710046271069943975, −7.42851745447157035217255184985, −6.45598557707379062984097466709, −4.47734364083323878121851964951, −2.34621893281690777347418491708, 1.95223962279921209108555583373, 4.64867329246519734774806254725, 5.37282860250868997087030688576, 7.23448601714033035807640724692, 9.056286200204738732857756821595, 9.539110333937468323522989855521, 10.75279357728365336570991991051, 11.50711804704980327683513882450, 13.04808990839602679496943139962, 14.15948516550802033148615834547

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