Properties

Degree $2$
Conductor $192$
Sign $-0.333 + 0.942i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 2.82i)3-s − 5.65i·5-s − 6·7-s + (−7.00 − 5.65i)9-s − 5.65i·11-s − 10·13-s + (16.0 + 5.65i)15-s − 22.6i·17-s − 2·19-s + (6 − 16.9i)21-s − 11.3i·23-s − 7.00·25-s + (23.0 − 14.1i)27-s − 16.9i·29-s − 22·31-s + ⋯
L(s)  = 1  + (−0.333 + 0.942i)3-s − 1.13i·5-s − 0.857·7-s + (−0.777 − 0.628i)9-s − 0.514i·11-s − 0.769·13-s + (1.06 + 0.377i)15-s − 1.33i·17-s − 0.105·19-s + (0.285 − 0.808i)21-s − 0.491i·23-s − 0.280·25-s + (0.851 − 0.523i)27-s − 0.585i·29-s − 0.709·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.333 + 0.942i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.333 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-0.333 + 0.942i$
Motivic weight: \(2\)
Character: $\chi_{192} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1),\ -0.333 + 0.942i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.351606 - 0.497246i\)
\(L(\frac12)\) \(\approx\) \(0.351606 - 0.497246i\)
\(L(2)\) 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 \( 1 + (1 - 2.82i)T \)
good5 \( 1 + 5.65iT - 25T^{2} \)
7 \( 1 + 6T + 49T^{2} \)
11 \( 1 + 5.65iT - 121T^{2} \)
13 \( 1 + 10T + 169T^{2} \)
17 \( 1 + 22.6iT - 289T^{2} \)
19 \( 1 + 2T + 361T^{2} \)
23 \( 1 + 11.3iT - 529T^{2} \)
29 \( 1 + 16.9iT - 841T^{2} \)
31 \( 1 + 22T + 961T^{2} \)
37 \( 1 - 6T + 1.36e3T^{2} \)
41 \( 1 - 33.9iT - 1.68e3T^{2} \)
43 \( 1 + 82T + 1.84e3T^{2} \)
47 \( 1 - 67.8iT - 2.20e3T^{2} \)
53 \( 1 - 62.2iT - 2.80e3T^{2} \)
59 \( 1 + 73.5iT - 3.48e3T^{2} \)
61 \( 1 - 86T + 3.72e3T^{2} \)
67 \( 1 + 2T + 4.48e3T^{2} \)
71 \( 1 + 124. iT - 5.04e3T^{2} \)
73 \( 1 - 82T + 5.32e3T^{2} \)
79 \( 1 - 10T + 6.24e3T^{2} \)
83 \( 1 - 73.5iT - 6.88e3T^{2} \)
89 \( 1 + 33.9iT - 7.92e3T^{2} \)
97 \( 1 + 94T + 9.40e3T^{2} \)
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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

−12.01950614117946585443493170285, −11.04149073009512687228756808992, −9.727433502065386965565122863579, −9.340386222186197727321680144535, −8.203195787830959945570876474767, −6.59387133494967527884977702011, −5.35194265588145884900693417579, −4.51940446466197251972109600854, −3.06375294348636723508846077756, −0.34949447096554236819794990158, 2.11069408651266285238567965592, 3.46766842693641246302661146971, 5.46015142984845819772878767787, 6.68819988717016081537271772800, 7.09979244414894395021587982846, 8.380862154553730997341495473527, 9.877753904053585087025537825215, 10.66351050982243380319133115332, 11.72469184887904304875096522314, 12.65584047590198450828423606930

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