↑

Properties

Label 2-96-32.19-c2-0-13
Degree $2$
Conductor $96$
Sign $0.676 + 0.736i$
Analytic cond. $2.61581$
Root an. cond. $1.61734$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.92 − 0.526i)2-s + (−1.60 − 0.662i)3-s + (3.44 − 2.03i)4-s + (3.51 − 1.45i)5-s + (−3.43 − 0.435i)6-s + (−1.77 − 1.77i)7-s + (5.57 − 5.73i)8-s + (2.12 + 2.12i)9-s + (6.01 − 4.65i)10-s + (−3.84 + 1.59i)11-s + (−6.86 + 0.969i)12-s + (5.91 + 2.45i)13-s + (−4.35 − 2.48i)14-s − 6.58·15-s + (7.73 − 14.0i)16-s + 10.3i·17-s + ⋯
L(s)  = 1  + (0.964 − 0.263i)2-s + (−0.533 − 0.220i)3-s + (0.861 − 0.508i)4-s + (0.702 − 0.291i)5-s + (−0.572 − 0.0726i)6-s + (−0.253 − 0.253i)7-s + (0.697 − 0.717i)8-s + (0.235 + 0.235i)9-s + (0.601 − 0.465i)10-s + (−0.349 + 0.144i)11-s + (−0.571 + 0.0807i)12-s + (0.455 + 0.188i)13-s + (−0.310 − 0.177i)14-s − 0.439·15-s + (0.483 − 0.875i)16-s + 0.606i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(96\)    =    \(2^{5} \cdot 3\)
Sign: $0.676 + 0.736i$
Analytic conductor: \(2.61581\)
Root analytic conductor: \(1.61734\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{96} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 96,\ (\ :1),\ 0.676 + 0.736i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.90129 - 0.834544i\)
\(L(\frac12)\) \(\approx\) \(1.90129 - 0.834544i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.92 + 0.526i)T \)
3 \( 1 + (1.60 + 0.662i)T \)
good5 \( 1 + (-3.51 + 1.45i)T + (17.6 - 17.6i)T^{2} \)
7 \( 1 + (1.77 + 1.77i)T + 49iT^{2} \)
11 \( 1 + (3.84 - 1.59i)T + (85.5 - 85.5i)T^{2} \)
13 \( 1 + (-5.91 - 2.45i)T + (119. + 119. i)T^{2} \)
17 \( 1 - 10.3iT - 289T^{2} \)
19 \( 1 + (2.86 - 6.92i)T + (-255. - 255. i)T^{2} \)
23 \( 1 + (29.6 - 29.6i)T - 529iT^{2} \)
29 \( 1 + (10.0 - 24.2i)T + (-594. - 594. i)T^{2} \)
31 \( 1 + 7.83iT - 961T^{2} \)
37 \( 1 + (-56.8 + 23.5i)T + (968. - 968. i)T^{2} \)
41 \( 1 + (18.3 + 18.3i)T + 1.68e3iT^{2} \)
43 \( 1 + (-54.2 + 22.4i)T + (1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 + 55.9T + 2.20e3T^{2} \)
53 \( 1 + (-22.5 - 54.3i)T + (-1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (43.1 + 104. i)T + (-2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (6.15 - 14.8i)T + (-2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (-52.0 - 21.5i)T + (3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (-56.0 - 56.0i)T + 5.04e3iT^{2} \)
73 \( 1 + (89.9 + 89.9i)T + 5.32e3iT^{2} \)
79 \( 1 + 17.3T + 6.24e3T^{2} \)
83 \( 1 + (-2.34 + 5.66i)T + (-4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (-82.3 + 82.3i)T - 7.92e3iT^{2} \)
97 \( 1 + 96.8T + 9.40e3T^{2} \)
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

−13.37636205415232653484914421694, −12.74399918822023272953246621556, −11.62426514115698283354452953851, −10.58334576752547928180611116014, −9.589440060858010018568153416574, −7.63286447030027745534688185845, −6.23369556642504969963854906654, −5.44487809325204227807099131969, −3.87707329608220117610590297261, −1.78810536235293134559983509636, 2.61425607972098930962071260511, 4.40275019142549972879999721983, 5.80160943169033628098079622754, 6.48693623230348239966137065169, 8.054014092002253346575121469980, 9.780603027038442149805035928442, 10.85452893486448598895261440817, 11.87302871891144856376619851320, 12.95067480749362422471446919465, 13.79944124199077074326254374799

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