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Properties

Label 2-192-192.35-c1-0-24
Degree $2$
Conductor $192$
Sign $0.761 + 0.648i$
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  + (1.40 + 0.119i)2-s + (0.816 − 1.52i)3-s + (1.97 + 0.337i)4-s + (−2.90 − 1.94i)5-s + (1.33 − 2.05i)6-s + (0.941 + 2.27i)7-s + (2.73 + 0.712i)8-s + (−1.66 − 2.49i)9-s + (−3.86 − 3.08i)10-s + (−0.119 − 0.602i)11-s + (2.12 − 2.73i)12-s + (3.02 + 4.52i)13-s + (1.05 + 3.31i)14-s + (−5.33 + 2.85i)15-s + (3.77 + 1.33i)16-s + (−2.86 + 2.86i)17-s + ⋯
L(s)  = 1  + (0.996 + 0.0847i)2-s + (0.471 − 0.882i)3-s + (0.985 + 0.168i)4-s + (−1.29 − 0.867i)5-s + (0.544 − 0.838i)6-s + (0.355 + 0.858i)7-s + (0.967 + 0.251i)8-s + (−0.555 − 0.831i)9-s + (−1.22 − 0.975i)10-s + (−0.0361 − 0.181i)11-s + (0.613 − 0.789i)12-s + (0.837 + 1.25i)13-s + (0.281 + 0.885i)14-s + (−1.37 + 0.736i)15-s + (0.942 + 0.333i)16-s + (−0.694 + 0.694i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 + 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.761 + 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.761 + 0.648i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1/2),\ 0.761 + 0.648i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91494 - 0.705003i\)
\(L(\frac12)\) \(\approx\) \(1.91494 - 0.705003i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 - 0.119i)T \)
3 \( 1 + (-0.816 + 1.52i)T \)
good5 \( 1 + (2.90 + 1.94i)T + (1.91 + 4.61i)T^{2} \)
7 \( 1 + (-0.941 - 2.27i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (0.119 + 0.602i)T + (-10.1 + 4.20i)T^{2} \)
13 \( 1 + (-3.02 - 4.52i)T + (-4.97 + 12.0i)T^{2} \)
17 \( 1 + (2.86 - 2.86i)T - 17iT^{2} \)
19 \( 1 + (3.42 + 5.12i)T + (-7.27 + 17.5i)T^{2} \)
23 \( 1 + (2.04 - 4.94i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-6.35 - 1.26i)T + (26.7 + 11.0i)T^{2} \)
31 \( 1 + 2.22T + 31T^{2} \)
37 \( 1 + (1.95 + 1.30i)T + (14.1 + 34.1i)T^{2} \)
41 \( 1 + (-1.92 - 0.797i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (1.05 + 5.27i)T + (-39.7 + 16.4i)T^{2} \)
47 \( 1 + (5.80 + 5.80i)T + 47iT^{2} \)
53 \( 1 + (2.62 - 0.521i)T + (48.9 - 20.2i)T^{2} \)
59 \( 1 + (-3.90 + 5.83i)T + (-22.5 - 54.5i)T^{2} \)
61 \( 1 + (10.9 + 2.17i)T + (56.3 + 23.3i)T^{2} \)
67 \( 1 + (0.594 - 2.98i)T + (-61.8 - 25.6i)T^{2} \)
71 \( 1 + (1.46 - 0.606i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (-7.41 - 3.07i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (8.48 + 8.48i)T + 79iT^{2} \)
83 \( 1 + (-2.70 + 1.80i)T + (31.7 - 76.6i)T^{2} \)
89 \( 1 + (0.774 - 0.320i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 + 4.44iT - 97T^{2} \)
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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

−12.49629542608305826354015764222, −11.70107981044472829509784205912, −11.22691165790944205235929280743, −8.795208181919457380964570952741, −8.425197452561663028666595609612, −7.19365848387178626945260737355, −6.17171911734172801490605304573, −4.70867214274779787396855388145, −3.62430797339218687103622268560, −1.90974202800909136689345231732, 2.90481200336249521078316238117, 3.88257393555964083106221481441, 4.62768573185974815478691421248, 6.30379872822278110024297379790, 7.61779371275382074481060014983, 8.253523242510140769462754504817, 10.33437013683813575700522384017, 10.69684837683408877272130854788, 11.50552684163984780404360123640, 12.69421381601345740708150645651

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