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Properties

Label 2-192-192.131-c1-0-29
Degree $2$
Conductor $192$
Sign $-0.936 + 0.350i$
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  + (0.743 − 1.20i)2-s + (−0.416 − 1.68i)3-s + (−0.894 − 1.78i)4-s + (−1.69 + 2.52i)5-s + (−2.33 − 0.748i)6-s + (−1.66 − 4.02i)7-s + (−2.81 − 0.253i)8-s + (−2.65 + 1.40i)9-s + (1.78 + 3.91i)10-s + (3.48 − 0.692i)11-s + (−2.63 + 2.24i)12-s + (3.08 − 2.06i)13-s + (−6.08 − 0.987i)14-s + (4.95 + 1.78i)15-s + (−2.39 + 3.20i)16-s + (2.62 − 2.62i)17-s + ⋯
L(s)  = 1  + (0.525 − 0.850i)2-s + (−0.240 − 0.970i)3-s + (−0.447 − 0.894i)4-s + (−0.755 + 1.13i)5-s + (−0.952 − 0.305i)6-s + (−0.630 − 1.52i)7-s + (−0.995 − 0.0896i)8-s + (−0.884 + 0.467i)9-s + (0.565 + 1.23i)10-s + (1.04 − 0.208i)11-s + (−0.760 + 0.649i)12-s + (0.856 − 0.572i)13-s + (−1.62 − 0.264i)14-s + (1.28 + 0.461i)15-s + (−0.599 + 0.800i)16-s + (0.636 − 0.636i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 + 0.350i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.936 + 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-0.936 + 0.350i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1/2),\ -0.936 + 0.350i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.191894 - 1.06062i\)
\(L(\frac12)\) \(\approx\) \(0.191894 - 1.06062i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.743 + 1.20i)T \)
3 \( 1 + (0.416 + 1.68i)T \)
good5 \( 1 + (1.69 - 2.52i)T + (-1.91 - 4.61i)T^{2} \)
7 \( 1 + (1.66 + 4.02i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (-3.48 + 0.692i)T + (10.1 - 4.20i)T^{2} \)
13 \( 1 + (-3.08 + 2.06i)T + (4.97 - 12.0i)T^{2} \)
17 \( 1 + (-2.62 + 2.62i)T - 17iT^{2} \)
19 \( 1 + (-2.52 + 1.68i)T + (7.27 - 17.5i)T^{2} \)
23 \( 1 + (1.33 - 3.23i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-0.240 + 1.20i)T + (-26.7 - 11.0i)T^{2} \)
31 \( 1 + 5.17T + 31T^{2} \)
37 \( 1 + (-2.19 + 3.29i)T + (-14.1 - 34.1i)T^{2} \)
41 \( 1 + (-2.03 - 0.842i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (-10.1 + 2.01i)T + (39.7 - 16.4i)T^{2} \)
47 \( 1 + (3.16 + 3.16i)T + 47iT^{2} \)
53 \( 1 + (-1.90 - 9.57i)T + (-48.9 + 20.2i)T^{2} \)
59 \( 1 + (8.20 + 5.48i)T + (22.5 + 54.5i)T^{2} \)
61 \( 1 + (1.36 - 6.84i)T + (-56.3 - 23.3i)T^{2} \)
67 \( 1 + (-3.78 - 0.753i)T + (61.8 + 25.6i)T^{2} \)
71 \( 1 + (-4.21 + 1.74i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (1.11 + 0.460i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (-2.41 - 2.41i)T + 79iT^{2} \)
83 \( 1 + (-2.81 - 4.21i)T + (-31.7 + 76.6i)T^{2} \)
89 \( 1 + (6.04 - 2.50i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 + 3.52iT - 97T^{2} \)
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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

−12.00912654396071304402636657904, −11.15202226371643199581688608484, −10.68471404293999197673842857326, −9.389347086949188948910081140027, −7.64767588837944552573180441761, −6.90797959997316037426251379167, −5.88049616081248076187254805888, −3.89536019485489307687322128558, −3.13351888686642526599658938727, −0.908582946584122562816537708938, 3.48506719151465369000273595180, 4.40439598918673594508006439494, 5.57323657718042527095172927868, 6.33285777060265082879619991113, 8.185549180672492788497888291169, 8.941325877139937326433665071475, 9.473171911989806647747598871100, 11.43732102369169294174894781870, 12.20875988216415707231839212021, 12.66724972853708081946339298006

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