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Properties

Label 2-91-91.76-c1-0-5
Degree $2$
Conductor $91$
Sign $0.792 + 0.609i$
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  + (0.218 − 0.813i)2-s + (1.24 − 0.716i)3-s + (1.11 + 0.645i)4-s + (−1.01 + 1.01i)5-s + (−0.312 − 1.16i)6-s + (−2.62 + 0.330i)7-s + (1.95 − 1.95i)8-s + (−0.472 + 0.819i)9-s + (0.603 + 1.04i)10-s + (−4.68 − 1.25i)11-s + 1.84·12-s + (1.04 − 3.44i)13-s + (−0.303 + 2.20i)14-s + (−0.531 + 1.98i)15-s + (0.122 + 0.212i)16-s + (1.49 − 2.58i)17-s + ⋯
L(s)  = 1  + (0.154 − 0.575i)2-s + (0.716 − 0.413i)3-s + (0.558 + 0.322i)4-s + (−0.453 + 0.453i)5-s + (−0.127 − 0.476i)6-s + (−0.992 + 0.124i)7-s + (0.692 − 0.692i)8-s + (−0.157 + 0.273i)9-s + (0.190 + 0.330i)10-s + (−1.41 − 0.378i)11-s + 0.533·12-s + (0.291 − 0.956i)13-s + (−0.0811 + 0.590i)14-s + (−0.137 + 0.512i)15-s + (0.0306 + 0.0531i)16-s + (0.361 − 0.626i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 + 0.609i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.792 + 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.792 + 0.609i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.792 + 0.609i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17633 - 0.399819i\)
\(L(\frac12)\) \(\approx\) \(1.17633 - 0.399819i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.62 - 0.330i)T \)
13 \( 1 + (-1.04 + 3.44i)T \)
good2 \( 1 + (-0.218 + 0.813i)T + (-1.73 - i)T^{2} \)
3 \( 1 + (-1.24 + 0.716i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.01 - 1.01i)T - 5iT^{2} \)
11 \( 1 + (4.68 + 1.25i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.49 + 2.58i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.62 - 6.06i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.02 - 0.590i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.77 + 4.81i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.79 + 4.79i)T - 31iT^{2} \)
37 \( 1 + (-5.40 - 1.44i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-4.71 - 1.26i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (2.90 + 1.67i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.12 + 4.12i)T + 47iT^{2} \)
53 \( 1 - 5.79T + 53T^{2} \)
59 \( 1 + (10.7 - 2.88i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (7.95 + 4.59i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.508 - 1.89i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (3.19 - 0.855i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (0.353 + 0.353i)T + 73iT^{2} \)
79 \( 1 + 6.95T + 79T^{2} \)
83 \( 1 + (-3.22 + 3.22i)T - 83iT^{2} \)
89 \( 1 + (0.0636 - 0.237i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-2.43 - 9.08i)T + (-84.0 + 48.5i)T^{2} \)
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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

−13.52302636463119996637409218924, −13.07394643828091358195226037735, −11.93194453921970213119418148888, −10.80314596847613655633932334133, −9.889237664703703655078333166377, −7.980079534458477568368571032960, −7.56561161354368648020047547845, −5.86282680322614213519819546611, −3.40522409881095362454332563182, −2.66232335963218383038460533904, 2.86383582371628489735877461630, 4.59839347346849130706635164657, 6.16897917092614901150107472299, 7.37379558885917911156104239856, 8.579129965023901516593207218282, 9.733689592351341095082945332373, 10.84706870042610590351272460892, 12.20801190912075421813940907495, 13.37086995872229221515285262683, 14.40133472578334382217969848731

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