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Properties

Label 2-192-64.45-c1-0-5
Degree $2$
Conductor $192$
Sign $-0.567 - 0.823i$
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  + (0.479 + 1.33i)2-s + (0.555 + 0.831i)3-s + (−1.53 + 1.27i)4-s + (0.248 + 1.25i)5-s + (−0.839 + 1.13i)6-s + (0.0439 + 0.106i)7-s + (−2.43 − 1.43i)8-s + (−0.382 + 0.923i)9-s + (−1.54 + 0.931i)10-s + (0.502 + 0.335i)11-s + (−1.91 − 0.571i)12-s + (0.581 − 2.92i)13-s + (−0.120 + 0.109i)14-s + (−0.901 + 0.901i)15-s + (0.741 − 3.93i)16-s + (4.49 + 4.49i)17-s + ⋯
L(s)  = 1  + (0.339 + 0.940i)2-s + (0.320 + 0.480i)3-s + (−0.769 + 0.638i)4-s + (0.111 + 0.559i)5-s + (−0.342 + 0.464i)6-s + (0.0166 + 0.0401i)7-s + (−0.861 − 0.507i)8-s + (−0.127 + 0.307i)9-s + (−0.488 + 0.294i)10-s + (0.151 + 0.101i)11-s + (−0.553 − 0.164i)12-s + (0.161 − 0.811i)13-s + (−0.0320 + 0.0292i)14-s + (−0.232 + 0.232i)15-s + (0.185 − 0.982i)16-s + (1.09 + 1.09i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.567 - 0.823i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.567 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-0.567 - 0.823i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1/2),\ -0.567 - 0.823i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.655930 + 1.24948i\)
\(L(\frac12)\) \(\approx\) \(0.655930 + 1.24948i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.479 - 1.33i)T \)
3 \( 1 + (-0.555 - 0.831i)T \)
good5 \( 1 + (-0.248 - 1.25i)T + (-4.61 + 1.91i)T^{2} \)
7 \( 1 + (-0.0439 - 0.106i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (-0.502 - 0.335i)T + (4.20 + 10.1i)T^{2} \)
13 \( 1 + (-0.581 + 2.92i)T + (-12.0 - 4.97i)T^{2} \)
17 \( 1 + (-4.49 - 4.49i)T + 17iT^{2} \)
19 \( 1 + (3.56 + 0.709i)T + (17.5 + 7.27i)T^{2} \)
23 \( 1 + (-0.433 - 0.179i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (-4.44 + 2.97i)T + (11.0 - 26.7i)T^{2} \)
31 \( 1 + 3.80iT - 31T^{2} \)
37 \( 1 + (5.55 - 1.10i)T + (34.1 - 14.1i)T^{2} \)
41 \( 1 + (-7.52 - 3.11i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (1.41 - 2.11i)T + (-16.4 - 39.7i)T^{2} \)
47 \( 1 + (5.15 + 5.15i)T + 47iT^{2} \)
53 \( 1 + (-1.52 - 1.02i)T + (20.2 + 48.9i)T^{2} \)
59 \( 1 + (-1.59 - 8.03i)T + (-54.5 + 22.5i)T^{2} \)
61 \( 1 + (5.13 + 7.68i)T + (-23.3 + 56.3i)T^{2} \)
67 \( 1 + (2.61 + 3.92i)T + (-25.6 + 61.8i)T^{2} \)
71 \( 1 + (4.48 + 10.8i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (-2.49 + 6.02i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (9.80 - 9.80i)T - 79iT^{2} \)
83 \( 1 + (10.9 + 2.17i)T + (76.6 + 31.7i)T^{2} \)
89 \( 1 + (16.0 - 6.62i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 - 10.7iT - 97T^{2} \)
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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

−13.08079994275698877732914687880, −12.17373910674826892282457048934, −10.68804935868280745051263249599, −9.865698101527733145270594181545, −8.601006557196533494362058900123, −7.85115290060113550038257723022, −6.57052445458205911353257969095, −5.58433650949237207234886108679, −4.23305582271488705829746538912, −3.04123809789925565630464536242, 1.33373615139919717352728762814, 2.94048215429814927182518453261, 4.38568446758196856802198409093, 5.58371094187470172762539405845, 6.99173780730289900509812145451, 8.529117206062875024493681800973, 9.200428829725605860189071183904, 10.30481232724060176963569342061, 11.44948232416213376434785938882, 12.31683567348633121779675359588

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