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Properties

Label 2-91-91.88-c1-0-5
Degree $2$
Conductor $91$
Sign $0.901 - 0.432i$
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  + (1.16 + 0.672i)2-s + 2.05·3-s + (−0.0951 − 0.164i)4-s + (−3.08 + 1.78i)5-s + (2.38 + 1.37i)6-s + (−2.09 − 1.61i)7-s − 2.94i·8-s + 1.20·9-s − 4.79·10-s + 1.27i·11-s + (−0.195 − 0.337i)12-s + (3.57 − 0.474i)13-s + (−1.35 − 3.29i)14-s + (−6.33 + 3.65i)15-s + (1.79 − 3.10i)16-s + (3.86 + 6.70i)17-s + ⋯
L(s)  = 1  + (0.823 + 0.475i)2-s + 1.18·3-s + (−0.0475 − 0.0824i)4-s + (−1.38 + 0.797i)5-s + (0.975 + 0.562i)6-s + (−0.792 − 0.610i)7-s − 1.04i·8-s + 0.400·9-s − 1.51·10-s + 0.385i·11-s + (−0.0563 − 0.0975i)12-s + (0.991 − 0.131i)13-s + (−0.362 − 0.879i)14-s + (−1.63 + 0.944i)15-s + (0.447 − 0.775i)16-s + (0.938 + 1.62i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.901 - 0.432i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.901 - 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.901 - 0.432i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (88, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.901 - 0.432i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46612 + 0.333125i\)
\(L(\frac12)\) \(\approx\) \(1.46612 + 0.333125i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.09 + 1.61i)T \)
13 \( 1 + (-3.57 + 0.474i)T \)
good2 \( 1 + (-1.16 - 0.672i)T + (1 + 1.73i)T^{2} \)
3 \( 1 - 2.05T + 3T^{2} \)
5 \( 1 + (3.08 - 1.78i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 - 1.27iT - 11T^{2} \)
17 \( 1 + (-3.86 - 6.70i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 0.943iT - 19T^{2} \)
23 \( 1 + (-0.823 + 1.42i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.02 + 3.50i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.46 + 2.57i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.914 - 0.528i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.63 - 2.09i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.91 + 3.31i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.774 - 0.447i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.0399 + 0.0692i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-9.68 + 5.59i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 7.62T + 61T^{2} \)
67 \( 1 + 6.32iT - 67T^{2} \)
71 \( 1 + (-9.89 - 5.71i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.658 + 0.380i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.42 - 2.47i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.32iT - 83T^{2} \)
89 \( 1 + (-6.56 - 3.78i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.414 + 0.239i)T + (48.5 + 84.0i)T^{2} \)
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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

−14.37911823261153418143920042642, −13.38341229598286040705416368967, −12.49663899700131743100055881741, −10.89041379773308852619921482890, −9.804395724627502674925133001729, −8.280724550519056899889983303318, −7.34604438994938879851877755158, −6.15656917267311346462160669493, −3.87985133480666659032886850622, −3.52402416173741689451857261000, 3.05951869597916636657661077941, 3.74983675455719592028284606979, 5.29250502960440440883052509967, 7.55046102327394829464748385659, 8.587975007327748535678690780150, 9.175754005970484620671660803061, 11.33045577514534050856121359116, 12.08736356661332530832406783162, 13.01559425013423086961246392005, 13.80946343455740264438295827608

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