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Properties

Label 2-91-91.20-c1-0-5
Degree $2$
Conductor $91$
Sign $0.992 + 0.123i$
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  + (1.61 − 0.433i)2-s + (0.552 + 0.318i)3-s + (0.700 − 0.404i)4-s + (−1.42 + 1.42i)5-s + (1.03 + 0.276i)6-s + (0.234 − 2.63i)7-s + (−1.41 + 1.41i)8-s + (−1.29 − 2.24i)9-s + (−1.68 + 2.91i)10-s + (0.254 + 0.948i)11-s + 0.516·12-s + (−1.60 − 3.22i)13-s + (−0.764 − 4.36i)14-s + (−1.23 + 0.331i)15-s + (−2.48 + 4.29i)16-s + (2.99 + 5.18i)17-s + ⋯
L(s)  = 1  + (1.14 − 0.306i)2-s + (0.318 + 0.184i)3-s + (0.350 − 0.202i)4-s + (−0.635 + 0.635i)5-s + (0.421 + 0.112i)6-s + (0.0885 − 0.996i)7-s + (−0.498 + 0.498i)8-s + (−0.432 − 0.748i)9-s + (−0.532 + 0.922i)10-s + (0.0766 + 0.285i)11-s + 0.148·12-s + (−0.446 − 0.894i)13-s + (−0.204 − 1.16i)14-s + (−0.319 + 0.0856i)15-s + (−0.620 + 1.07i)16-s + (0.725 + 1.25i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.123i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.123i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.992 + 0.123i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.992 + 0.123i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52741 - 0.0945397i\)
\(L(\frac12)\) \(\approx\) \(1.52741 - 0.0945397i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.234 + 2.63i)T \)
13 \( 1 + (1.60 + 3.22i)T \)
good2 \( 1 + (-1.61 + 0.433i)T + (1.73 - i)T^{2} \)
3 \( 1 + (-0.552 - 0.318i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.42 - 1.42i)T - 5iT^{2} \)
11 \( 1 + (-0.254 - 0.948i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-2.99 - 5.18i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.71 - 0.726i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-2.58 - 1.49i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.65 + 6.33i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.11 - 6.11i)T - 31iT^{2} \)
37 \( 1 + (1.24 + 4.63i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (0.886 + 3.30i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (0.748 - 0.432i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.17 + 2.17i)T + 47iT^{2} \)
53 \( 1 - 6.32T + 53T^{2} \)
59 \( 1 + (0.131 - 0.491i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-11.2 + 6.51i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.827 - 0.221i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (3.01 - 11.2i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (1.03 + 1.03i)T + 73iT^{2} \)
79 \( 1 + 11.6T + 79T^{2} \)
83 \( 1 + (1.23 - 1.23i)T - 83iT^{2} \)
89 \( 1 + (7.75 - 2.07i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (12.0 + 3.23i)T + (84.0 + 48.5i)T^{2} \)
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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

−14.25300699239834826324600457606, −13.02789110030740180250508546413, −12.12980448026414884248053970330, −11.10322625834076227628781216432, −9.992775105750254326112290096171, −8.361058394636697506876816596863, −7.13369956964334260910473222757, −5.55963558479828584076454992496, −3.93787529166917997336311442676, −3.23169668049578743910642172289, 2.94213577786834742887138155156, 4.69994601156879211238078707989, 5.49745431580916611468590664137, 7.13073801365915847537753558068, 8.508529155907356766179258251744, 9.451729164272011955276278364808, 11.53178201570029792972294686172, 12.13809061346947485165398405874, 13.19430822938024563661601104731, 14.16475784492221226749162471843

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