↑

Properties

Label 2-91-91.51-c1-0-0
Degree $2$
Conductor $91$
Sign $0.922 - 0.385i$
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  + (−0.929 − 0.536i)2-s + (1.21 + 2.10i)3-s + (−0.424 − 0.734i)4-s + (0.541 + 0.312i)5-s − 2.60i·6-s + (2.34 + 1.21i)7-s + 3.05i·8-s + (−1.45 + 2.52i)9-s + (−0.335 − 0.581i)10-s + (−0.613 + 0.354i)11-s + (1.03 − 1.78i)12-s + (0.848 − 3.50i)13-s + (−1.53 − 2.39i)14-s + 1.52i·15-s + (0.791 − 1.37i)16-s + (−1.67 − 2.89i)17-s + ⋯
L(s)  = 1  + (−0.657 − 0.379i)2-s + (0.701 + 1.21i)3-s + (−0.212 − 0.367i)4-s + (0.242 + 0.139i)5-s − 1.06i·6-s + (0.888 + 0.459i)7-s + 1.08i·8-s + (−0.485 + 0.840i)9-s + (−0.106 − 0.183i)10-s + (−0.185 + 0.106i)11-s + (0.297 − 0.515i)12-s + (0.235 − 0.971i)13-s + (−0.409 − 0.638i)14-s + 0.392i·15-s + (0.197 − 0.342i)16-s + (−0.405 − 0.702i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.385i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.922 - 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.922 - 0.385i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.922 - 0.385i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.866463 + 0.173663i\)
\(L(\frac12)\) \(\approx\) \(0.866463 + 0.173663i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.34 - 1.21i)T \)
13 \( 1 + (-0.848 + 3.50i)T \)
good2 \( 1 + (0.929 + 0.536i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-1.21 - 2.10i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.541 - 0.312i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.613 - 0.354i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.67 + 2.89i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.50 + 2.60i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.21 - 3.83i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.59T + 29T^{2} \)
31 \( 1 + (-3.80 + 2.19i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.366 + 0.211i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 5.01iT - 41T^{2} \)
43 \( 1 - 11.2T + 43T^{2} \)
47 \( 1 + (6.99 + 4.03i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.348 - 0.603i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-8.54 + 4.93i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.34 - 4.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.02 - 5.21i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 14.0iT - 71T^{2} \)
73 \( 1 + (4.40 - 2.54i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.95 + 3.38i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.2iT - 83T^{2} \)
89 \( 1 + (-11.5 - 6.68i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.202iT - 97T^{2} \)
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

−14.48161824863592306715297304665, −13.35847632260112077411351582578, −11.51212239808598100438667527446, −10.62321265690933101193849556298, −9.775155368452811041179035812224, −8.894154837952367796641388971325, −8.023765587431898354649048384878, −5.62108612676731222687781827847, −4.44151668219954097927642427719, −2.45617401562077076141043778453, 1.81181873650907394438582298479, 4.11818090296131782691924359164, 6.42966156960714116187380308346, 7.53449086104323600136500656970, 8.275406811112421405413525971683, 9.103612252720291020051980708767, 10.70008035524334243250847834858, 12.19915783578700963354780321028, 13.12096848693716332986123776688, 13.87038517321903808416074222408

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