↑

Properties

Label 2-91-91.6-c1-0-2
Degree $2$
Conductor $91$
Sign $0.927 - 0.372i$
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

Related objects

Downloads

Learn more

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 + (2.43 + 1.02i)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 + (0.921 + 0.388i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.372i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 - 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.927 - 0.372i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.927 - 0.372i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03290 + 0.199691i\)
\(L(\frac12)\) \(\approx\) \(1.03290 + 0.199691i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.43 - 1.02i)T \)
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 \( 1 + (2.43 - 9.08i)T + (-84.0 - 48.5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.55243160433627207147803887295, −13.09554497179421092910963129132, −11.97217324207169146448268721528, −10.94935305987438357295947841797, −10.15694152137106116828959511140, −8.147331005758559507434560410435, −7.18050552087846032322130261585, −5.85756942106038820207181770810, −5.25612324869529437969997395281, −2.28621558111997335157873377981, 2.20852673932104709761642285869, 4.40697062306992403344645205162, 5.48609023512459293403130543435, 7.17916011003458386151542202349, 8.468135391084492660503096559746, 10.14546138080649793418234982491, 11.05958472052096884170383951235, 11.51052392330103968641390937290, 12.95986311807488377064779741415, 13.65211434158465534544663499952

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