Properties

Label 2-192-192.11-c1-0-28
Degree $2$
Conductor $192$
Sign $-0.722 + 0.691i$
Analytic cond. $1.53312$
Root an. cond. $1.23819$
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  + (0.640 − 1.26i)2-s + (−1.16 − 1.28i)3-s + (−1.17 − 1.61i)4-s + (3.57 − 2.39i)5-s + (−2.36 + 0.647i)6-s + (−0.994 + 2.40i)7-s + (−2.79 + 0.450i)8-s + (−0.286 + 2.98i)9-s + (−0.721 − 6.04i)10-s + (0.714 − 3.59i)11-s + (−0.698 + 3.39i)12-s + (−1.38 + 2.08i)13-s + (2.38 + 2.79i)14-s + (−7.23 − 1.80i)15-s + (−1.22 + 3.80i)16-s + (0.951 + 0.951i)17-s + ⋯
L(s)  = 1  + (0.453 − 0.891i)2-s + (−0.672 − 0.740i)3-s + (−0.589 − 0.807i)4-s + (1.60 − 1.06i)5-s + (−0.964 + 0.264i)6-s + (−0.375 + 0.907i)7-s + (−0.987 + 0.159i)8-s + (−0.0954 + 0.995i)9-s + (−0.228 − 1.91i)10-s + (0.215 − 1.08i)11-s + (−0.201 + 0.979i)12-s + (−0.385 + 0.576i)13-s + (0.638 + 0.746i)14-s + (−1.86 − 0.465i)15-s + (−0.305 + 0.952i)16-s + (0.230 + 0.230i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.722 + 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-0.722 + 0.691i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1/2),\ -0.722 + 0.691i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.496105 - 1.23654i\)
\(L(\frac12)\) \(\approx\) \(0.496105 - 1.23654i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.640 + 1.26i)T \)
3 \( 1 + (1.16 + 1.28i)T \)
good5 \( 1 + (-3.57 + 2.39i)T + (1.91 - 4.61i)T^{2} \)
7 \( 1 + (0.994 - 2.40i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (-0.714 + 3.59i)T + (-10.1 - 4.20i)T^{2} \)
13 \( 1 + (1.38 - 2.08i)T + (-4.97 - 12.0i)T^{2} \)
17 \( 1 + (-0.951 - 0.951i)T + 17iT^{2} \)
19 \( 1 + (0.214 - 0.321i)T + (-7.27 - 17.5i)T^{2} \)
23 \( 1 + (-1.14 - 2.75i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (-2.53 + 0.504i)T + (26.7 - 11.0i)T^{2} \)
31 \( 1 - 2.03T + 31T^{2} \)
37 \( 1 + (-4.28 + 2.86i)T + (14.1 - 34.1i)T^{2} \)
41 \( 1 + (4.29 - 1.77i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (1.88 - 9.45i)T + (-39.7 - 16.4i)T^{2} \)
47 \( 1 + (-5.57 + 5.57i)T - 47iT^{2} \)
53 \( 1 + (1.47 + 0.292i)T + (48.9 + 20.2i)T^{2} \)
59 \( 1 + (3.66 + 5.48i)T + (-22.5 + 54.5i)T^{2} \)
61 \( 1 + (4.09 - 0.814i)T + (56.3 - 23.3i)T^{2} \)
67 \( 1 + (-1.50 - 7.58i)T + (-61.8 + 25.6i)T^{2} \)
71 \( 1 + (6.27 + 2.59i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (15.4 - 6.40i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (-2.31 + 2.31i)T - 79iT^{2} \)
83 \( 1 + (-1.62 - 1.08i)T + (31.7 + 76.6i)T^{2} \)
89 \( 1 + (4.04 + 1.67i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 - 1.86iT - 97T^{2} \)
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

−12.26930846194378025049381879097, −11.51803176043533026396146918524, −10.25394386435064004707193361633, −9.326748583670651292175140522548, −8.545647110712087984032097210186, −6.25485149432710859252033539011, −5.78749371977300999319610129286, −4.84958248584707164455854618182, −2.55226506570359365362785425053, −1.31279714822121033333017076487, 3.02809619142792949514395390427, 4.53238685468070760162666962480, 5.64796675514152972785521981976, 6.59049010778400343746164415349, 7.23546691638817212238319961208, 9.214546836811049695338556575856, 10.05147269552662879003997611874, 10.55289674979731478342506200575, 12.11239797560528910579301892567, 13.16240438309049435454913354072

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