↑

Properties

Label 2-91-91.16-c1-0-3
Degree $2$
Conductor $91$
Sign $-0.0420 + 0.999i$
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  − 2.38·2-s + (1.37 − 2.38i)3-s + 3.70·4-s + (−0.491 + 0.850i)5-s + (−3.28 + 5.69i)6-s + (−0.911 − 2.48i)7-s − 4.06·8-s + (−2.28 − 3.95i)9-s + (1.17 − 2.03i)10-s + (0.293 − 0.509i)11-s + (5.09 − 8.82i)12-s + (2.39 − 2.69i)13-s + (2.17 + 5.93i)14-s + (1.35 + 2.34i)15-s + 2.30·16-s − 6.45·17-s + ⋯
L(s)  = 1  − 1.68·2-s + (0.794 − 1.37i)3-s + 1.85·4-s + (−0.219 + 0.380i)5-s + (−1.34 + 2.32i)6-s + (−0.344 − 0.938i)7-s − 1.43·8-s + (−0.761 − 1.31i)9-s + (0.370 − 0.642i)10-s + (0.0886 − 0.153i)11-s + (1.47 − 2.54i)12-s + (0.663 − 0.748i)13-s + (0.581 + 1.58i)14-s + (0.348 + 0.604i)15-s + 0.576·16-s − 1.56·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0420 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0420 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.0420 + 0.999i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ -0.0420 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.387090 - 0.403726i\)
\(L(\frac12)\) \(\approx\) \(0.387090 - 0.403726i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.911 + 2.48i)T \)
13 \( 1 + (-2.39 + 2.69i)T \)
good2 \( 1 + 2.38T + 2T^{2} \)
3 \( 1 + (-1.37 + 2.38i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.491 - 0.850i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.293 + 0.509i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 6.45T + 17T^{2} \)
19 \( 1 + (-1.91 - 3.31i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 8.26T + 23T^{2} \)
29 \( 1 + (-1.98 - 3.42i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.49 - 2.58i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.75T + 37T^{2} \)
41 \( 1 + (1.83 + 3.17i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.19 - 5.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.17 + 3.75i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.212 + 0.368i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 6.00T + 59T^{2} \)
61 \( 1 + (1.10 + 1.91i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.50 - 6.07i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.80 - 3.11i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.46 + 4.27i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.39 - 2.41i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.86T + 83T^{2} \)
89 \( 1 + 2.09T + 89T^{2} \)
97 \( 1 + (3.84 - 6.66i)T + (-48.5 - 84.0i)T^{2} \)
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

−13.63455827510047640744937011154, −12.88651406517246008285243002962, −11.30736858741022161026689467936, −10.47615387251609870953260983200, −9.042986889757231182177571062684, −8.242858608775841862859623617953, −7.19482148657063242044390288357, −6.69450342940659265769873821881, −3.03834500414905677830030191429, −1.20050738829685340082387120652, 2.64012804798322466310884390192, 4.58230946543916308580631230010, 6.70848710077142416325594022033, 8.453081858981078757719166546161, 8.974136893670771329719701190971, 9.530295632716137851986006853048, 10.76055129157585605263440856908, 11.61851090641497564946169244080, 13.40517790828564310338933454750, 15.03856824599576471663234253522

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