↑

Properties

Label 2-96-96.11-c1-0-4
Degree $2$
Conductor $96$
Sign $0.994 + 0.104i$
Analytic cond. $0.766563$
Root an. cond. $0.875536$
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  + (−1.41 − 0.101i)2-s + (1.72 − 0.0979i)3-s + (1.97 + 0.285i)4-s + (−0.180 − 0.0746i)5-s + (−2.44 − 0.0366i)6-s + (−0.289 + 0.289i)7-s + (−2.76 − 0.602i)8-s + (2.98 − 0.338i)9-s + (0.246 + 0.123i)10-s + (3.10 + 1.28i)11-s + (3.45 + 0.299i)12-s + (−1.27 − 3.08i)13-s + (0.436 − 0.378i)14-s + (−0.318 − 0.111i)15-s + (3.83 + 1.12i)16-s − 0.806·17-s + ⋯
L(s)  = 1  + (−0.997 − 0.0714i)2-s + (0.998 − 0.0565i)3-s + (0.989 + 0.142i)4-s + (−0.0805 − 0.0333i)5-s + (−0.999 − 0.0149i)6-s + (−0.109 + 0.109i)7-s + (−0.977 − 0.212i)8-s + (0.993 − 0.112i)9-s + (0.0779 + 0.0390i)10-s + (0.937 + 0.388i)11-s + (0.996 + 0.0863i)12-s + (−0.354 − 0.856i)13-s + (0.116 − 0.101i)14-s + (−0.0823 − 0.0287i)15-s + (0.959 + 0.282i)16-s − 0.195·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.104i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.104i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(96\)    =    \(2^{5} \cdot 3\)
Sign: $0.994 + 0.104i$
Analytic conductor: \(0.766563\)
Root analytic conductor: \(0.875536\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{96} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 96,\ (\ :1/2),\ 0.994 + 0.104i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.883099 - 0.0462402i\)
\(L(\frac12)\) \(\approx\) \(0.883099 - 0.0462402i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.101i)T \)
3 \( 1 + (-1.72 + 0.0979i)T \)
good5 \( 1 + (0.180 + 0.0746i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (0.289 - 0.289i)T - 7iT^{2} \)
11 \( 1 + (-3.10 - 1.28i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (1.27 + 3.08i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 + 0.806T + 17T^{2} \)
19 \( 1 + (5.57 - 2.31i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (5.03 - 5.03i)T - 23iT^{2} \)
29 \( 1 + (3.64 + 8.78i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 - 7.48iT - 31T^{2} \)
37 \( 1 + (1.47 - 3.55i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-2.62 - 2.62i)T + 41iT^{2} \)
43 \( 1 + (-2.84 + 6.87i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + 0.399iT - 47T^{2} \)
53 \( 1 + (-1.42 + 3.43i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-0.918 + 2.21i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-8.81 + 3.64i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (3.76 + 9.09i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (-2.86 - 2.86i)T + 71iT^{2} \)
73 \( 1 + (6.97 - 6.97i)T - 73iT^{2} \)
79 \( 1 - 8.01T + 79T^{2} \)
83 \( 1 + (-2.47 - 5.96i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-5.43 + 5.43i)T - 89iT^{2} \)
97 \( 1 - 2.61T + 97T^{2} \)
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

−14.14593191932188733824354219783, −12.74344854986909866810193667764, −11.83745287242802044608458220803, −10.30461218286117718773798629287, −9.557238549173719577628672453763, −8.442475466736745192650038050814, −7.60085036330407610186406931183, −6.28860871203508447151995955919, −3.81435664424896482369143589609, −2.08448976738010764482467064738, 2.12835496573373350391972835100, 3.96816197899860388669551934299, 6.41402420189062051858111545491, 7.45281702244034972225113995766, 8.720723598439207371591131010715, 9.314633329782978123669358711389, 10.51533397354735444831145290819, 11.67089222947747176019092494306, 12.94040899809908871821687585135, 14.34732001191896174791109411793

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