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Properties

Label 2-192-192.35-c1-0-16
Degree $2$
Conductor $192$
Sign $0.0332 + 0.999i$
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  + (−0.216 + 1.39i)2-s + (−1.36 + 1.06i)3-s + (−1.90 − 0.604i)4-s + (−2.38 − 1.59i)5-s + (−1.18 − 2.14i)6-s + (0.537 + 1.29i)7-s + (1.25 − 2.53i)8-s + (0.746 − 2.90i)9-s + (2.74 − 2.98i)10-s + (−0.120 − 0.607i)11-s + (3.25 − 1.19i)12-s + (−2.88 − 4.32i)13-s + (−1.92 + 0.470i)14-s + (4.95 − 0.349i)15-s + (3.26 + 2.30i)16-s + (−5.60 + 5.60i)17-s + ⋯
L(s)  = 1  + (−0.152 + 0.988i)2-s + (−0.790 + 0.612i)3-s + (−0.953 − 0.302i)4-s + (−1.06 − 0.711i)5-s + (−0.484 − 0.874i)6-s + (0.203 + 0.490i)7-s + (0.444 − 0.895i)8-s + (0.248 − 0.968i)9-s + (0.866 − 0.944i)10-s + (−0.0364 − 0.183i)11-s + (0.938 − 0.345i)12-s + (−0.800 − 1.19i)13-s + (−0.515 + 0.125i)14-s + (1.27 − 0.0903i)15-s + (0.817 + 0.576i)16-s + (−1.35 + 1.35i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0332 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0332 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.0332 + 0.999i$
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: \((2,\ 192,\ (\ :1/2),\ 0.0332 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0776459 - 0.0751074i\)
\(L(\frac12)\) \(\approx\) \(0.0776459 - 0.0751074i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.216 - 1.39i)T \)
3 \( 1 + (1.36 - 1.06i)T \)
good5 \( 1 + (2.38 + 1.59i)T + (1.91 + 4.61i)T^{2} \)
7 \( 1 + (-0.537 - 1.29i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (0.120 + 0.607i)T + (-10.1 + 4.20i)T^{2} \)
13 \( 1 + (2.88 + 4.32i)T + (-4.97 + 12.0i)T^{2} \)
17 \( 1 + (5.60 - 5.60i)T - 17iT^{2} \)
19 \( 1 + (2.11 + 3.16i)T + (-7.27 + 17.5i)T^{2} \)
23 \( 1 + (-0.461 + 1.11i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (1.62 + 0.323i)T + (26.7 + 11.0i)T^{2} \)
31 \( 1 + 5.33T + 31T^{2} \)
37 \( 1 + (6.92 + 4.62i)T + (14.1 + 34.1i)T^{2} \)
41 \( 1 + (3.36 + 1.39i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (-2.45 - 12.3i)T + (-39.7 + 16.4i)T^{2} \)
47 \( 1 + (-0.576 - 0.576i)T + 47iT^{2} \)
53 \( 1 + (-6.93 + 1.38i)T + (48.9 - 20.2i)T^{2} \)
59 \( 1 + (1.57 - 2.36i)T + (-22.5 - 54.5i)T^{2} \)
61 \( 1 + (-3.21 - 0.640i)T + (56.3 + 23.3i)T^{2} \)
67 \( 1 + (1.03 - 5.19i)T + (-61.8 - 25.6i)T^{2} \)
71 \( 1 + (-4.85 + 2.01i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (-2.50 - 1.03i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (6.29 + 6.29i)T + 79iT^{2} \)
83 \( 1 + (10.0 - 6.68i)T + (31.7 - 76.6i)T^{2} \)
89 \( 1 + (2.13 - 0.883i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 + 1.28iT - 97T^{2} \)
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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

−12.50807617983806299651469833409, −11.23433881718125574579816090805, −10.31721142848919809300407966399, −8.975673948955693934856695914079, −8.354390576065775466406657317623, −7.11695359485651770332029669611, −5.82768348079643414113701364635, −4.88290814196167883775582554590, −3.97758001219088390699015480633, −0.10820084710758744802377999759, 2.14967964315614211346950142316, 3.94598680863160061602300156976, 4.97214353673144239878210341199, 6.95540593931469225340451036617, 7.46869597146930158176142584091, 8.871977254665550358383768974253, 10.23476676450963029057320761808, 11.12568471213324399360125871828, 11.66489656057266692437204991002, 12.35401492218456928335433983735

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