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Properties

Label 2-192-192.107-c1-0-13
Degree $2$
Conductor $192$
Sign $0.927 - 0.373i$
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  + (−0.486 + 1.32i)2-s + (0.609 − 1.62i)3-s + (−1.52 − 1.29i)4-s + (1.96 + 2.93i)5-s + (1.85 + 1.59i)6-s + (1.41 − 3.41i)7-s + (2.45 − 1.40i)8-s + (−2.25 − 1.97i)9-s + (−4.85 + 1.17i)10-s + (−1.35 − 0.269i)11-s + (−3.02 + 1.68i)12-s + (3.69 + 2.47i)13-s + (3.84 + 3.53i)14-s + (5.96 − 1.39i)15-s + (0.666 + 3.94i)16-s + (1.07 + 1.07i)17-s + ⋯
L(s)  = 1  + (−0.343 + 0.939i)2-s + (0.351 − 0.936i)3-s + (−0.763 − 0.645i)4-s + (0.877 + 1.31i)5-s + (0.758 + 0.652i)6-s + (0.534 − 1.28i)7-s + (0.868 − 0.495i)8-s + (−0.752 − 0.658i)9-s + (−1.53 + 0.372i)10-s + (−0.408 − 0.0812i)11-s + (−0.872 + 0.487i)12-s + (1.02 + 0.685i)13-s + (1.02 + 0.944i)14-s + (1.53 − 0.359i)15-s + (0.166 + 0.986i)16-s + (0.261 + 0.261i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.927 - 0.373i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1/2),\ 0.927 - 0.373i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19576 + 0.231568i\)
\(L(\frac12)\) \(\approx\) \(1.19576 + 0.231568i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.486 - 1.32i)T \)
3 \( 1 + (-0.609 + 1.62i)T \)
good5 \( 1 + (-1.96 - 2.93i)T + (-1.91 + 4.61i)T^{2} \)
7 \( 1 + (-1.41 + 3.41i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (1.35 + 0.269i)T + (10.1 + 4.20i)T^{2} \)
13 \( 1 + (-3.69 - 2.47i)T + (4.97 + 12.0i)T^{2} \)
17 \( 1 + (-1.07 - 1.07i)T + 17iT^{2} \)
19 \( 1 + (-2.55 - 1.70i)T + (7.27 + 17.5i)T^{2} \)
23 \( 1 + (-0.348 - 0.842i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (1.64 + 8.26i)T + (-26.7 + 11.0i)T^{2} \)
31 \( 1 + 7.50T + 31T^{2} \)
37 \( 1 + (1.83 + 2.74i)T + (-14.1 + 34.1i)T^{2} \)
41 \( 1 + (9.24 - 3.82i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (2.87 + 0.571i)T + (39.7 + 16.4i)T^{2} \)
47 \( 1 + (3.91 - 3.91i)T - 47iT^{2} \)
53 \( 1 + (-0.750 + 3.77i)T + (-48.9 - 20.2i)T^{2} \)
59 \( 1 + (1.97 - 1.32i)T + (22.5 - 54.5i)T^{2} \)
61 \( 1 + (-2.15 - 10.8i)T + (-56.3 + 23.3i)T^{2} \)
67 \( 1 + (-1.15 + 0.230i)T + (61.8 - 25.6i)T^{2} \)
71 \( 1 + (-7.50 - 3.10i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (3.51 - 1.45i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (-8.03 + 8.03i)T - 79iT^{2} \)
83 \( 1 + (2.45 - 3.66i)T + (-31.7 - 76.6i)T^{2} \)
89 \( 1 + (0.100 + 0.0416i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 - 12.5iT - 97T^{2} \)
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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

−13.30797981754706917203034926537, −11.35886180398016619916392205956, −10.50313692298028305959651102806, −9.552492522119343295674129283405, −8.192782121937937197820666149885, −7.34627095541711291647175436250, −6.62875266098296997929488465932, −5.70705976089018636271583919928, −3.69151066825994470323822296162, −1.63802476184764792203359818324, 1.83471809343033544764785780661, 3.31633765636661813554415391150, 5.15505435406441959614917167223, 5.29606998698956024604424007639, 8.182487373914830262994451719912, 8.820148612797256427993935141370, 9.359580333218615154125634450744, 10.40861822708489183501704527727, 11.39135185450417587845494570878, 12.45597778206081412164232320703

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