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Properties

Label 2-192-192.107-c1-0-18
Degree $2$
Conductor $192$
Sign $0.689 + 0.724i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.484 − 1.32i)2-s + (1.73 − 0.0238i)3-s + (−1.52 − 1.28i)4-s + (2.22 + 3.33i)5-s + (0.808 − 2.31i)6-s + (0.140 − 0.338i)7-s + (−2.45 + 1.40i)8-s + (2.99 − 0.0824i)9-s + (5.51 − 1.34i)10-s + (−4.16 − 0.827i)11-s + (−2.67 − 2.19i)12-s + (−1.81 − 1.21i)13-s + (−0.381 − 0.350i)14-s + (3.93 + 5.72i)15-s + (0.679 + 3.94i)16-s + (−2.34 − 2.34i)17-s + ⋯
L(s)  = 1  + (0.342 − 0.939i)2-s + (0.999 − 0.0137i)3-s + (−0.764 − 0.644i)4-s + (0.996 + 1.49i)5-s + (0.329 − 0.943i)6-s + (0.0530 − 0.127i)7-s + (−0.867 + 0.497i)8-s + (0.999 − 0.0274i)9-s + (1.74 − 0.424i)10-s + (−1.25 − 0.249i)11-s + (−0.773 − 0.633i)12-s + (−0.502 − 0.335i)13-s + (−0.102 − 0.0936i)14-s + (1.01 + 1.47i)15-s + (0.169 + 0.985i)16-s + (−0.568 − 0.568i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.689 + 0.724i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.689 + 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.689 + 0.724i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1/2),\ 0.689 + 0.724i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67980 - 0.720315i\)
\(L(\frac12)\) \(\approx\) \(1.67980 - 0.720315i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.484 + 1.32i)T \)
3 \( 1 + (-1.73 + 0.0238i)T \)
good5 \( 1 + (-2.22 - 3.33i)T + (-1.91 + 4.61i)T^{2} \)
7 \( 1 + (-0.140 + 0.338i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (4.16 + 0.827i)T + (10.1 + 4.20i)T^{2} \)
13 \( 1 + (1.81 + 1.21i)T + (4.97 + 12.0i)T^{2} \)
17 \( 1 + (2.34 + 2.34i)T + 17iT^{2} \)
19 \( 1 + (5.58 + 3.73i)T + (7.27 + 17.5i)T^{2} \)
23 \( 1 + (0.561 + 1.35i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (0.511 + 2.57i)T + (-26.7 + 11.0i)T^{2} \)
31 \( 1 - 7.17T + 31T^{2} \)
37 \( 1 + (-2.65 - 3.96i)T + (-14.1 + 34.1i)T^{2} \)
41 \( 1 + (-5.19 + 2.15i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (0.439 + 0.0874i)T + (39.7 + 16.4i)T^{2} \)
47 \( 1 + (-2.23 + 2.23i)T - 47iT^{2} \)
53 \( 1 + (1.03 - 5.20i)T + (-48.9 - 20.2i)T^{2} \)
59 \( 1 + (6.44 - 4.30i)T + (22.5 - 54.5i)T^{2} \)
61 \( 1 + (-2.26 - 11.3i)T + (-56.3 + 23.3i)T^{2} \)
67 \( 1 + (-3.80 + 0.757i)T + (61.8 - 25.6i)T^{2} \)
71 \( 1 + (-3.54 - 1.46i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (8.70 - 3.60i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (-2.50 + 2.50i)T - 79iT^{2} \)
83 \( 1 + (1.65 - 2.47i)T + (-31.7 - 76.6i)T^{2} \)
89 \( 1 + (11.1 + 4.63i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 + 10.0iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.70549717322733672457793871584, −11.12807248059424609925385572842, −10.39235434549231049516294157434, −9.826631502382829440501215344120, −8.647483960289427484667246262036, −7.28255463612415746927107528277, −6.04642154430400808162827726089, −4.50357328707296818519171700658, −2.73943228914962099846244533380, −2.49877522311618626904913577777, 2.21466933438216299270468024900, 4.25635429342559960006271556901, 5.12313065274748319707376705801, 6.32480885820092394930267005383, 7.86153785198855842254889938196, 8.498005998433998872069486716695, 9.345796175626186133444176418786, 10.19887757661727426869586901391, 12.51869327159977588739699310665, 12.83063867094100162113746026184

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