Properties

Label 2-192-64.45-c1-0-2
Degree $2$
Conductor $192$
Sign $-0.625 - 0.779i$
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  + (−0.899 + 1.09i)2-s + (0.555 + 0.831i)3-s + (−0.382 − 1.96i)4-s + (0.713 + 3.58i)5-s + (−1.40 − 0.141i)6-s + (−0.499 − 1.20i)7-s + (2.48 + 1.34i)8-s + (−0.382 + 0.923i)9-s + (−4.55 − 2.44i)10-s + (0.165 + 0.110i)11-s + (1.41 − 1.40i)12-s + (−0.763 + 3.83i)13-s + (1.76 + 0.539i)14-s + (−2.58 + 2.58i)15-s + (−3.70 + 1.50i)16-s + (−3.40 − 3.40i)17-s + ⋯
L(s)  = 1  + (−0.635 + 0.771i)2-s + (0.320 + 0.480i)3-s + (−0.191 − 0.981i)4-s + (0.319 + 1.60i)5-s + (−0.574 − 0.0576i)6-s + (−0.188 − 0.456i)7-s + (0.879 + 0.476i)8-s + (−0.127 + 0.307i)9-s + (−1.44 − 0.773i)10-s + (0.0497 + 0.0332i)11-s + (0.409 − 0.406i)12-s + (−0.211 + 1.06i)13-s + (0.472 + 0.144i)14-s + (−0.667 + 0.667i)15-s + (−0.926 + 0.375i)16-s + (−0.824 − 0.824i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.625 - 0.779i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.625 - 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-0.625 - 0.779i$
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: \((2,\ 192,\ (\ :1/2),\ -0.625 - 0.779i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.400865 + 0.835544i\)
\(L(\frac12)\) \(\approx\) \(0.400865 + 0.835544i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.899 - 1.09i)T \)
3 \( 1 + (-0.555 - 0.831i)T \)
good5 \( 1 + (-0.713 - 3.58i)T + (-4.61 + 1.91i)T^{2} \)
7 \( 1 + (0.499 + 1.20i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (-0.165 - 0.110i)T + (4.20 + 10.1i)T^{2} \)
13 \( 1 + (0.763 - 3.83i)T + (-12.0 - 4.97i)T^{2} \)
17 \( 1 + (3.40 + 3.40i)T + 17iT^{2} \)
19 \( 1 + (-1.60 - 0.319i)T + (17.5 + 7.27i)T^{2} \)
23 \( 1 + (-5.45 - 2.25i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (4.85 - 3.24i)T + (11.0 - 26.7i)T^{2} \)
31 \( 1 + 7.27iT - 31T^{2} \)
37 \( 1 + (-10.9 + 2.18i)T + (34.1 - 14.1i)T^{2} \)
41 \( 1 + (-4.05 - 1.67i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (-4.87 + 7.30i)T + (-16.4 - 39.7i)T^{2} \)
47 \( 1 + (-7.45 - 7.45i)T + 47iT^{2} \)
53 \( 1 + (-6.21 - 4.15i)T + (20.2 + 48.9i)T^{2} \)
59 \( 1 + (1.52 + 7.68i)T + (-54.5 + 22.5i)T^{2} \)
61 \( 1 + (0.775 + 1.16i)T + (-23.3 + 56.3i)T^{2} \)
67 \( 1 + (2.61 + 3.91i)T + (-25.6 + 61.8i)T^{2} \)
71 \( 1 + (3.70 + 8.95i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (0.717 - 1.73i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (7.84 - 7.84i)T - 79iT^{2} \)
83 \( 1 + (8.03 + 1.59i)T + (76.6 + 31.7i)T^{2} \)
89 \( 1 + (9.80 - 4.06i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 - 6.53iT - 97T^{2} \)
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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

−13.44277325064472241494314841429, −11.25706219156803627864221505223, −10.86185039401310161998674962333, −9.644422403971298346313980519030, −9.205343227374405316456305698600, −7.45757036000761819743043897171, −7.00332258945355786122062542556, −5.83494172895362497856173496813, −4.21881920925903999976686063069, −2.50156277704827264716036050042, 1.06336801348087981768907239312, 2.62651117447409197108138122103, 4.36481501149676709477500036381, 5.73772252612494056676031170861, 7.46129115364995220396535020177, 8.592045352529709412847775716417, 8.964415172179109565114609890728, 10.03545372828784477569139063366, 11.29740685220005056843844759637, 12.46784662144360177888438280272

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