↑

Properties

Label 2-91-13.9-c1-0-6
Degree $2$
Conductor $91$
Sign $0.189 + 0.981i$
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  + (0.866 − 1.5i)2-s + (0.366 − 0.633i)3-s + (−0.5 − 0.866i)4-s − 1.73·5-s + (−0.633 − 1.09i)6-s + (−0.5 − 0.866i)7-s + 1.73·8-s + (1.23 + 2.13i)9-s + (−1.49 + 2.59i)10-s + (−2.36 + 4.09i)11-s − 0.732·12-s + (−1.59 − 3.23i)13-s − 1.73·14-s + (−0.633 + 1.09i)15-s + (2.49 − 4.33i)16-s + (2.13 + 3.69i)17-s + ⋯
L(s)  = 1  + (0.612 − 1.06i)2-s + (0.211 − 0.366i)3-s + (−0.250 − 0.433i)4-s − 0.774·5-s + (−0.258 − 0.448i)6-s + (−0.188 − 0.327i)7-s + 0.612·8-s + (0.410 + 0.711i)9-s + (−0.474 + 0.821i)10-s + (−0.713 + 1.23i)11-s − 0.211·12-s + (−0.443 − 0.896i)13-s − 0.462·14-s + (−0.163 + 0.283i)15-s + (0.624 − 1.08i)16-s + (0.517 + 0.896i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.189 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.189 + 0.981i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.189 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.990672 - 0.817482i\)
\(L(\frac12)\) \(\approx\) \(0.990672 - 0.817482i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (1.59 + 3.23i)T \)
good2 \( 1 + (-0.866 + 1.5i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.366 + 0.633i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 1.73T + 5T^{2} \)
11 \( 1 + (2.36 - 4.09i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.13 - 3.69i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.633 - 1.09i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.19T + 31T^{2} \)
37 \( 1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.59 - 4.5i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.09 + 8.83i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 0.928T + 47T^{2} \)
53 \( 1 - 3.92T + 53T^{2} \)
59 \( 1 + (5.36 + 9.29i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.59 - 13.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.09 - 3.63i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 7.19T + 73T^{2} \)
79 \( 1 - 5.80T + 79T^{2} \)
83 \( 1 - 8.19T + 83T^{2} \)
89 \( 1 + (-0.464 + 0.803i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.19 - 12.4i)T + (-48.5 + 84.0i)T^{2} \)
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

−13.32536340167557721342427635463, −12.80316833138061168411988829780, −11.98981485345654387406345439731, −10.69464968726998806775468811745, −10.04778944953097759754507823259, −7.907149105955935916610388692921, −7.36707189308310396421472832899, −5.02394607171507602611155640516, −3.79992588545711933676505352967, −2.20239267430026053557451768584, 3.52957957793634842374956169025, 4.87308718222571916877411924801, 6.20603939431763398514667040850, 7.36913073524405301302245032399, 8.489321419830709194583118338272, 9.848580526336600023353608529743, 11.24416137813438625121988054174, 12.38054621005926265129480445177, 13.63268696961314826551065625080, 14.54923954186050177500622965526

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