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Properties

Label 2-91-91.25-c1-0-3
Degree $2$
Conductor $91$
Sign $0.725 - 0.687i$
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  + (−1.84 + 1.06i)2-s + (0.0894 − 0.154i)3-s + (1.25 − 2.18i)4-s + (3.12 − 1.80i)5-s + 0.380i·6-s + (−1.20 + 2.35i)7-s + 1.10i·8-s + (1.48 + 2.57i)9-s + (−3.83 + 6.63i)10-s + (3.45 + 1.99i)11-s + (−0.225 − 0.389i)12-s + (−2.51 − 2.58i)13-s + (−0.274 − 5.61i)14-s − 0.644i·15-s + (1.34 + 2.33i)16-s + (2.39 − 4.14i)17-s + ⋯
L(s)  = 1  + (−1.30 + 0.751i)2-s + (0.0516 − 0.0894i)3-s + (0.629 − 1.09i)4-s + (1.39 − 0.806i)5-s + 0.155i·6-s + (−0.457 + 0.889i)7-s + 0.389i·8-s + (0.494 + 0.856i)9-s + (−1.21 + 2.09i)10-s + (1.04 + 0.601i)11-s + (−0.0649 − 0.112i)12-s + (−0.698 − 0.715i)13-s + (−0.0734 − 1.50i)14-s − 0.166i·15-s + (0.337 + 0.583i)16-s + (0.580 − 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.725 - 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.725 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.725 - 0.687i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.725 - 0.687i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.612732 + 0.244207i\)
\(L(\frac12)\) \(\approx\) \(0.612732 + 0.244207i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (1.20 - 2.35i)T \)
13 \( 1 + (2.51 + 2.58i)T \)
good2 \( 1 + (1.84 - 1.06i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-0.0894 + 0.154i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-3.12 + 1.80i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.45 - 1.99i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.39 + 4.14i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.72 - 1.57i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.08 + 1.88i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.57T + 29T^{2} \)
31 \( 1 + (1.28 + 0.743i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.29 - 2.48i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.11iT - 41T^{2} \)
43 \( 1 + 1.43T + 43T^{2} \)
47 \( 1 + (0.882 - 0.509i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.01 - 5.22i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.24 + 2.45i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.01 - 1.76i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.38 + 1.95i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.80iT - 71T^{2} \)
73 \( 1 + (-2.67 - 1.54i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.984 + 1.70i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.66iT - 83T^{2} \)
89 \( 1 + (-11.0 + 6.39i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.35iT - 97T^{2} \)
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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

−14.43220721470661057712172168383, −13.09975648704417748513164148622, −12.29866846342312096083651902132, −10.26575057773635245458946678884, −9.565580272298623974391979216417, −8.908762612417814331517389350583, −7.59454722222392282441876403191, −6.31291709295714839012282770288, −5.18172163762859265146384383811, −1.88202770229691155814512467317, 1.69598772035101159798568284621, 3.54575165461559176440033136269, 6.20541103340616928723532461626, 7.17694468674694202104770265600, 9.006462443755232010282653093071, 9.726468790446003095808358800835, 10.35784059588994108982980718737, 11.38861075850204996375986151582, 12.74351702762822507406367548345, 14.04212975521038480069498777590

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