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Properties

Label 2-192-12.11-c7-0-17
Degree $2$
Conductor $192$
Sign $0.223 - 0.974i$
Analytic cond. $59.9779$
Root an. cond. $7.74454$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (45.5 + 10.4i)3-s + 183. i·5-s − 1.23e3i·7-s + (1.96e3 + 951. i)9-s − 7.56e3·11-s + 3.59e3·13-s + (−1.91e3 + 8.37e3i)15-s + 1.07e4i·17-s + 3.96e4i·19-s + (1.29e4 − 5.64e4i)21-s + 3.50e4·23-s + 4.43e4·25-s + (7.98e4 + 6.39e4i)27-s + 1.34e5i·29-s − 2.52e4i·31-s + ⋯
L(s)  = 1  + (0.974 + 0.223i)3-s + 0.657i·5-s − 1.36i·7-s + (0.900 + 0.434i)9-s − 1.71·11-s + 0.454·13-s + (−0.146 + 0.640i)15-s + 0.530i·17-s + 1.32i·19-s + (0.304 − 1.33i)21-s + 0.600·23-s + 0.568·25-s + (0.780 + 0.624i)27-s + 1.02i·29-s − 0.152i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.223 - 0.974i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.223 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.223 - 0.974i$
Analytic conductor: \(59.9779\)
Root analytic conductor: \(7.74454\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :7/2),\ 0.223 - 0.974i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.574903841\)
\(L(\frac12)\) \(\approx\) \(2.574903841\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-45.5 - 10.4i)T \)
good5 \( 1 - 183. iT - 7.81e4T^{2} \)
7 \( 1 + 1.23e3iT - 8.23e5T^{2} \)
11 \( 1 + 7.56e3T + 1.94e7T^{2} \)
13 \( 1 - 3.59e3T + 6.27e7T^{2} \)
17 \( 1 - 1.07e4iT - 4.10e8T^{2} \)
19 \( 1 - 3.96e4iT - 8.93e8T^{2} \)
23 \( 1 - 3.50e4T + 3.40e9T^{2} \)
29 \( 1 - 1.34e5iT - 1.72e10T^{2} \)
31 \( 1 + 2.52e4iT - 2.75e10T^{2} \)
37 \( 1 - 5.16e5T + 9.49e10T^{2} \)
41 \( 1 - 3.74e5iT - 1.94e11T^{2} \)
43 \( 1 - 1.00e6iT - 2.71e11T^{2} \)
47 \( 1 + 7.96e5T + 5.06e11T^{2} \)
53 \( 1 + 5.59e5iT - 1.17e12T^{2} \)
59 \( 1 + 1.04e6T + 2.48e12T^{2} \)
61 \( 1 - 2.18e6T + 3.14e12T^{2} \)
67 \( 1 + 1.85e6iT - 6.06e12T^{2} \)
71 \( 1 + 8.66e5T + 9.09e12T^{2} \)
73 \( 1 - 1.15e6T + 1.10e13T^{2} \)
79 \( 1 + 1.47e6iT - 1.92e13T^{2} \)
83 \( 1 - 1.90e6T + 2.71e13T^{2} \)
89 \( 1 - 5.33e6iT - 4.42e13T^{2} \)
97 \( 1 + 1.95e6T + 8.07e13T^{2} \)
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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

−10.96019428055090202697169096469, −10.49664589709868557009942783199, −9.681641633711103035004347515885, −8.113573123648903528852221802528, −7.69653224088369649120094167817, −6.51665402051528348703701160580, −4.84847722114093549464402544144, −3.65593848697698941142061678422, −2.77306443031208620063783689136, −1.26131719169943034024685331926, 0.58507267091435989299864137581, 2.25945534190559243138148812317, 2.91426362813995476465103014781, 4.67853595761657237568140709440, 5.62306352714206359176018638566, 7.14222027347983924086680843074, 8.281603618904967261012747047070, 8.848455783229994949243732661236, 9.734695790569454150582179299292, 11.07663376303714925543570434570

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