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Properties

Label 2-192-16.11-c10-0-38
Degree $2$
Conductor $192$
Sign $-0.636 + 0.771i$
Analytic cond. $121.988$
Root an. cond. $11.0448$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−99.2 − 99.2i)3-s + (3.24e3 + 3.24e3i)5-s + 1.96e3·7-s + 1.96e4i·9-s + (1.91e5 − 1.91e5i)11-s + (1.62e5 − 1.62e5i)13-s − 6.43e5i·15-s − 1.30e6·17-s + (−2.29e6 − 2.29e6i)19-s + (−1.94e5 − 1.94e5i)21-s − 4.18e6·23-s + 1.12e7i·25-s + (1.95e6 − 1.95e6i)27-s + (−2.96e6 + 2.96e6i)29-s + 1.94e7i·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (1.03 + 1.03i)5-s + 0.116·7-s + 0.333i·9-s + (1.19 − 1.19i)11-s + (0.436 − 0.436i)13-s − 0.847i·15-s − 0.917·17-s + (−0.928 − 0.928i)19-s + (−0.0477 − 0.0477i)21-s − 0.650·23-s + 1.15i·25-s + (0.136 − 0.136i)27-s + (−0.144 + 0.144i)29-s + 0.679i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.636 + 0.771i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.636 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-0.636 + 0.771i$
Analytic conductor: \(121.988\)
Root analytic conductor: \(11.0448\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :5),\ -0.636 + 0.771i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.255755270\)
\(L(\frac12)\) \(\approx\) \(1.255755270\)
\(L(6)\) 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 \( 1 + (99.2 + 99.2i)T \)
good5 \( 1 + (-3.24e3 - 3.24e3i)T + 9.76e6iT^{2} \)
7 \( 1 - 1.96e3T + 2.82e8T^{2} \)
11 \( 1 + (-1.91e5 + 1.91e5i)T - 2.59e10iT^{2} \)
13 \( 1 + (-1.62e5 + 1.62e5i)T - 1.37e11iT^{2} \)
17 \( 1 + 1.30e6T + 2.01e12T^{2} \)
19 \( 1 + (2.29e6 + 2.29e6i)T + 6.13e12iT^{2} \)
23 \( 1 + 4.18e6T + 4.14e13T^{2} \)
29 \( 1 + (2.96e6 - 2.96e6i)T - 4.20e14iT^{2} \)
31 \( 1 - 1.94e7iT - 8.19e14T^{2} \)
37 \( 1 + (7.51e7 + 7.51e7i)T + 4.80e15iT^{2} \)
41 \( 1 - 5.96e7iT - 1.34e16T^{2} \)
43 \( 1 + (-1.35e8 + 1.35e8i)T - 2.16e16iT^{2} \)
47 \( 1 + 2.44e8iT - 5.25e16T^{2} \)
53 \( 1 + (1.52e7 + 1.52e7i)T + 1.74e17iT^{2} \)
59 \( 1 + (8.04e8 - 8.04e8i)T - 5.11e17iT^{2} \)
61 \( 1 + (-4.23e8 + 4.23e8i)T - 7.13e17iT^{2} \)
67 \( 1 + (4.90e8 + 4.90e8i)T + 1.82e18iT^{2} \)
71 \( 1 - 3.15e9T + 3.25e18T^{2} \)
73 \( 1 - 1.58e8iT - 4.29e18T^{2} \)
79 \( 1 + 9.42e8iT - 9.46e18T^{2} \)
83 \( 1 + (3.47e8 + 3.47e8i)T + 1.55e19iT^{2} \)
89 \( 1 - 1.02e10iT - 3.11e19T^{2} \)
97 \( 1 + 1.65e10T + 7.37e19T^{2} \)
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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

−10.78514006802872722573916235290, −9.337394249935913369869058574103, −8.438396584391783624838984936437, −6.84289909748878498235687077222, −6.39847868217923852108449400345, −5.47451260093718932058417903078, −3.83770046714562559230188326988, −2.58219741512960728158185640510, −1.56428851157753320986730053654, −0.25178940116369520673524816329, 1.34786900547322450516929014415, 1.98628535854943647014066860616, 4.06330270518284604282045407825, 4.69297667090643271709561248985, 5.93719633846858825825826650781, 6.66444757056567983700199570737, 8.327780891526807656457891457612, 9.329240943897389574830146131433, 9.800675176915289600406792065862, 11.01560705350576019586725562594

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