↑

Properties

Label 2-91-13.4-c1-0-0
Degree $2$
Conductor $91$
Sign $0.999 - 0.0179i$
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  + (−2.34 − 1.35i)2-s + (0.172 − 0.299i)3-s + (2.65 + 4.59i)4-s + 3.25i·5-s + (−0.809 + 0.467i)6-s + (0.866 − 0.5i)7-s − 8.94i·8-s + (1.44 + 2.49i)9-s + (4.40 − 7.62i)10-s + (−1.59 − 0.923i)11-s + 1.83·12-s + (3.60 − 0.0186i)13-s − 2.70·14-s + (0.976 + 0.563i)15-s + (−6.77 + 11.7i)16-s + (1.07 + 1.86i)17-s + ⋯
L(s)  = 1  + (−1.65 − 0.955i)2-s + (0.0998 − 0.172i)3-s + (1.32 + 2.29i)4-s + 1.45i·5-s + (−0.330 + 0.190i)6-s + (0.327 − 0.188i)7-s − 3.16i·8-s + (0.480 + 0.831i)9-s + (1.39 − 2.41i)10-s + (−0.482 − 0.278i)11-s + 0.530·12-s + (0.999 − 0.00517i)13-s − 0.722·14-s + (0.252 + 0.145i)15-s + (−1.69 + 2.93i)16-s + (0.261 + 0.452i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0179i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0179i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.999 - 0.0179i$
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: \((2,\ 91,\ (\ :1/2),\ 0.999 - 0.0179i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.516786 + 0.00464996i\)
\(L(\frac12)\) \(\approx\) \(0.516786 + 0.00464996i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.60 + 0.0186i)T \)
good2 \( 1 + (2.34 + 1.35i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-0.172 + 0.299i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 3.25iT - 5T^{2} \)
11 \( 1 + (1.59 + 0.923i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.07 - 1.86i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.07 - 1.20i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.906 + 1.56i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.36 + 2.36i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.74iT - 31T^{2} \)
37 \( 1 + (5.14 + 2.96i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.65 - 2.11i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.34 + 7.51i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 5.87iT - 47T^{2} \)
53 \( 1 + 9.30T + 53T^{2} \)
59 \( 1 + (-9.31 + 5.37i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.05 + 8.75i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.716 - 0.413i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.03 - 1.17i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 3.19iT - 73T^{2} \)
79 \( 1 - 0.801T + 79T^{2} \)
83 \( 1 - 9.97iT - 83T^{2} \)
89 \( 1 + (-13.0 - 7.55i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.99 + 4.61i)T + (48.5 - 84.0i)T^{2} \)
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

−13.92802739253723968327507907221, −12.72445650472869823660388092692, −11.32302589332906685414204396186, −10.66953830070653899987936052629, −10.15779047926813955383381398830, −8.520966052539286673967887366064, −7.69624975273577739826657174995, −6.64619432212293665357357644329, −3.47338299881913156388781426731, −2.05450775482962959007259840822, 1.22293935163406960725887619243, 4.92815942746469885708727773451, 6.23837485752598112880882444687, 7.65850693232735462667520976752, 8.709413326472613263630172046941, 9.234550760857855467564719156606, 10.34453934857356329873789188470, 11.68225469549773789298871611416, 13.04813522200350616562662047122, 14.61320508355478812822099678685

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