Properties

Label 2-91-13.4-c1-0-4
Degree $2$
Conductor $91$
Sign $0.510 + 0.859i$
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  + (−0.104 − 0.0601i)2-s + (0.291 − 0.504i)3-s + (−0.992 − 1.71i)4-s − 1.68i·5-s + (−0.0606 + 0.0350i)6-s + (0.866 − 0.5i)7-s + 0.479i·8-s + (1.33 + 2.30i)9-s + (−0.101 + 0.175i)10-s + (−0.315 − 0.182i)11-s − 1.15·12-s + (1.80 + 3.12i)13-s − 0.120·14-s + (−0.851 − 0.491i)15-s + (−1.95 + 3.38i)16-s + (−1.59 − 2.75i)17-s + ⋯
L(s)  = 1  + (−0.0737 − 0.0425i)2-s + (0.168 − 0.291i)3-s + (−0.496 − 0.859i)4-s − 0.754i·5-s + (−0.0247 + 0.0143i)6-s + (0.327 − 0.188i)7-s + 0.169i·8-s + (0.443 + 0.768i)9-s + (−0.0321 + 0.0556i)10-s + (−0.0952 − 0.0549i)11-s − 0.333·12-s + (0.499 + 0.866i)13-s − 0.0321·14-s + (−0.219 − 0.126i)15-s + (−0.489 + 0.847i)16-s + (−0.386 − 0.669i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.510 + 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.510 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.510 + 0.859i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.510 + 0.859i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.820116 - 0.466775i\)
\(L(\frac12)\) \(\approx\) \(0.820116 - 0.466775i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (-1.80 - 3.12i)T \)
good2 \( 1 + (0.104 + 0.0601i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-0.291 + 0.504i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 1.68iT - 5T^{2} \)
11 \( 1 + (0.315 + 0.182i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.59 + 2.75i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.25 + 0.721i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.54 - 4.40i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.09 - 7.09i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.69iT - 31T^{2} \)
37 \( 1 + (-5.46 - 3.15i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.04 + 2.91i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.386 + 0.669i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 12.7iT - 47T^{2} \)
53 \( 1 - 1.37T + 53T^{2} \)
59 \( 1 + (8.10 - 4.68i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.51 - 7.81i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (11.6 + 6.73i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.13 - 3.54i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 2.16iT - 73T^{2} \)
79 \( 1 + 6.88T + 79T^{2} \)
83 \( 1 - 0.567iT - 83T^{2} \)
89 \( 1 + (0.986 + 0.569i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.86 + 3.96i)T + (48.5 - 84.0i)T^{2} \)
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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

−13.62151288858231215663084333055, −13.31554447817478519986734591303, −11.72575748788072295408765003407, −10.63374690430838140540302015221, −9.421765498431197751024017380089, −8.524443141005441307557740477070, −7.12332679942187281364256127086, −5.44558607206749497715105644866, −4.41979208773680325032077142203, −1.59748407297795819747856135109, 3.10046089155102883658439449909, 4.33903251128538769950282213286, 6.26361765052030902561084829094, 7.65815757533037800427439932755, 8.668367118625445863073796652441, 9.862788259486217118760893676671, 10.99957762739406084169328499667, 12.28885007567378894583645469973, 13.12279433044963635116242238379, 14.37763154604650423345099119513

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