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Properties

Label 2-192-24.5-c2-0-2
Degree $2$
Conductor $192$
Sign $-0.892 - 0.451i$
Analytic cond. $5.23162$
Root an. cond. $2.28727$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.23 + 2i)3-s − 7.74·5-s − 3.46·7-s + (1.00 + 8.94i)9-s − 13.4·11-s + 20.7i·13-s + (−17.3 − 15.4i)15-s + 4i·19-s + (−7.74 − 6.92i)21-s − 30.9i·23-s + 35.0·25-s + (−15.6 + 22.0i)27-s − 7.74·29-s + 24.2·31-s + (−30.0 − 26.8i)33-s + ⋯
L(s)  = 1  + (0.745 + 0.666i)3-s − 1.54·5-s − 0.494·7-s + (0.111 + 0.993i)9-s − 1.21·11-s + 1.59i·13-s + (−1.15 − 1.03i)15-s + 0.210i·19-s + (−0.368 − 0.329i)21-s − 1.34i·23-s + 1.40·25-s + (−0.579 + 0.814i)27-s − 0.267·29-s + 0.782·31-s + (−0.909 − 0.813i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.892 - 0.451i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.892 - 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-0.892 - 0.451i$
Analytic conductor: \(5.23162\)
Root analytic conductor: \(2.28727\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1),\ -0.892 - 0.451i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.173965 + 0.729941i\)
\(L(\frac12)\) \(\approx\) \(0.173965 + 0.729941i\)
\(L(2)\) 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 \( 1 + (-2.23 - 2i)T \)
good5 \( 1 + 7.74T + 25T^{2} \)
7 \( 1 + 3.46T + 49T^{2} \)
11 \( 1 + 13.4T + 121T^{2} \)
13 \( 1 - 20.7iT - 169T^{2} \)
17 \( 1 - 289T^{2} \)
19 \( 1 - 4iT - 361T^{2} \)
23 \( 1 + 30.9iT - 529T^{2} \)
29 \( 1 + 7.74T + 841T^{2} \)
31 \( 1 - 24.2T + 961T^{2} \)
37 \( 1 - 34.6iT - 1.36e3T^{2} \)
41 \( 1 - 53.6iT - 1.68e3T^{2} \)
43 \( 1 + 52iT - 1.84e3T^{2} \)
47 \( 1 - 61.9iT - 2.20e3T^{2} \)
53 \( 1 - 54.2T + 2.80e3T^{2} \)
59 \( 1 - 40.2T + 3.48e3T^{2} \)
61 \( 1 + 6.92iT - 3.72e3T^{2} \)
67 \( 1 + 28iT - 4.48e3T^{2} \)
71 \( 1 - 30.9iT - 5.04e3T^{2} \)
73 \( 1 - 74T + 5.32e3T^{2} \)
79 \( 1 - 51.9T + 6.24e3T^{2} \)
83 \( 1 + 120.T + 6.88e3T^{2} \)
89 \( 1 - 53.6iT - 7.92e3T^{2} \)
97 \( 1 + 62T + 9.40e3T^{2} \)
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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

−12.69628813455615806306148129583, −11.66326618326626748658614047792, −10.75344378276523522314652278111, −9.731007025718832800227762906253, −8.565800522822786692373937040218, −7.898039246780748598195127278240, −6.76125210068749883281429712007, −4.78718711575287386014846420144, −3.96103935843980726437905307765, −2.71502667583170812685042756647, 0.38569749428616685395493355647, 2.84773487435238986734590973301, 3.75011821125977321651615241000, 5.48731854788779466688616558412, 7.14436624875926694264349080945, 7.81820588874808568522802625562, 8.468759292198167614044645127050, 9.875895636357221364818707822807, 11.01974614230825976382252447804, 12.11583736479914122660257278979

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