↑

Properties

Label 2-97-97.91-c1-0-2
Degree $2$
Conductor $97$
Sign $-0.392 + 0.919i$
Analytic cond. $0.774548$
Root an. cond. $0.880084$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 + 0.866i)2-s + (−2.36 + 1.36i)3-s + (0.5 − 0.866i)4-s + (0.232 − 0.866i)5-s + (2.36 − 4.09i)6-s + (−1.36 − 0.366i)7-s − 1.73i·8-s + (2.23 − 3.86i)9-s + (0.401 + 1.5i)10-s + (−3 − 1.73i)11-s + 2.73i·12-s + (−0.133 + 0.5i)13-s + (2.36 − 0.633i)14-s + (0.633 + 2.36i)15-s + (2.49 + 4.33i)16-s + (−5.59 + 1.5i)17-s + ⋯
L(s)  = 1  + (−1.06 + 0.612i)2-s + (−1.36 + 0.788i)3-s + (0.250 − 0.433i)4-s + (0.103 − 0.387i)5-s + (0.965 − 1.67i)6-s + (−0.516 − 0.138i)7-s − 0.612i·8-s + (0.744 − 1.28i)9-s + (0.127 + 0.474i)10-s + (−0.904 − 0.522i)11-s + 0.788i·12-s + (−0.0371 + 0.138i)13-s + (0.632 − 0.169i)14-s + (0.163 + 0.610i)15-s + (0.624 + 1.08i)16-s + (−1.35 + 0.363i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.392 + 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.392 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(97\)
Sign: $-0.392 + 0.919i$
Analytic conductor: \(0.774548\)
Root analytic conductor: \(0.880084\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 97,\ (\ :1/2),\ -0.392 + 0.919i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad97 \( 1 + (9 - 4i)T \)
good2 \( 1 + (1.5 - 0.866i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (2.36 - 1.36i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.232 + 0.866i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (1.36 + 0.366i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (3 + 1.73i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.133 - 0.5i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (5.59 - 1.5i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.267 + 0.267i)T + 19iT^{2} \)
23 \( 1 + (8.19 - 2.19i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-2.36 + 8.83i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (5.36 - 3.09i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.96 + 1.06i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (1.90 - 7.09i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-1.09 - 1.90i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 12.9T + 47T^{2} \)
53 \( 1 + (2.59 - 1.5i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.73 + 1.26i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.866 + 1.5i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.732 - 0.732i)T + 67iT^{2} \)
71 \( 1 + (-3.63 - 13.5i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-3.19 - 5.53i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 6iT - 79T^{2} \)
83 \( 1 + (-5.36 + 1.43i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 - 6.46iT - 89T^{2} \)
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

−13.46385496431320786322316976404, −12.42051326620279157439174217098, −11.09834396210556976769138996886, −10.25149301931082784039782435856, −9.376351778942095276987034592989, −8.167546737691063710064921623745, −6.66313573760057585099069640377, −5.68453019213349724916531252503, −4.18385563070461793963473214298, 0, 2.20707893917941011764918952367, 5.15562357750306937119228483213, 6.43271010030582409414494630439, 7.53147628535938555598608560693, 8.998776468956634929590956354801, 10.44779908660801688124636768210, 10.77128648557267204800907376992, 12.03351563046322045793083113324, 12.71738870939138273622607947798

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