Properties

Label 2-192-48.29-c2-0-1
Degree $2$
Conductor $192$
Sign $-0.416 - 0.909i$
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  + (−1.13 + 2.77i)3-s + (6.28 + 6.28i)5-s + 1.64i·7-s + (−6.43 − 6.29i)9-s + (4.75 + 4.75i)11-s + (−9.35 − 9.35i)13-s + (−24.5 + 10.3i)15-s + 11.4i·17-s + (8.58 + 8.58i)19-s + (−4.57 − 1.86i)21-s − 16.2·23-s + 54.0i·25-s + (24.7 − 10.7i)27-s + (−10.7 + 10.7i)29-s − 6.35·31-s + ⋯
L(s)  = 1  + (−0.377 + 0.926i)3-s + (1.25 + 1.25i)5-s + 0.235i·7-s + (−0.715 − 0.699i)9-s + (0.432 + 0.432i)11-s + (−0.719 − 0.719i)13-s + (−1.63 + 0.689i)15-s + 0.675i·17-s + (0.451 + 0.451i)19-s + (−0.217 − 0.0887i)21-s − 0.706·23-s + 2.16i·25-s + (0.917 − 0.398i)27-s + (−0.370 + 0.370i)29-s − 0.204·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.416 - 0.909i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.416 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-0.416 - 0.909i$
Analytic conductor: \(5.23162\)
Root analytic conductor: \(2.28727\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1),\ -0.416 - 0.909i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.806900 + 1.25734i\)
\(L(\frac12)\) \(\approx\) \(0.806900 + 1.25734i\)
\(L(2)\) 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 \( 1 + (1.13 - 2.77i)T \)
good5 \( 1 + (-6.28 - 6.28i)T + 25iT^{2} \)
7 \( 1 - 1.64iT - 49T^{2} \)
11 \( 1 + (-4.75 - 4.75i)T + 121iT^{2} \)
13 \( 1 + (9.35 + 9.35i)T + 169iT^{2} \)
17 \( 1 - 11.4iT - 289T^{2} \)
19 \( 1 + (-8.58 - 8.58i)T + 361iT^{2} \)
23 \( 1 + 16.2T + 529T^{2} \)
29 \( 1 + (10.7 - 10.7i)T - 841iT^{2} \)
31 \( 1 + 6.35T + 961T^{2} \)
37 \( 1 + (-27.2 + 27.2i)T - 1.36e3iT^{2} \)
41 \( 1 - 1.98T + 1.68e3T^{2} \)
43 \( 1 + (-19.4 + 19.4i)T - 1.84e3iT^{2} \)
47 \( 1 + 74.9iT - 2.20e3T^{2} \)
53 \( 1 + (-4.00 - 4.00i)T + 2.80e3iT^{2} \)
59 \( 1 + (-27.9 - 27.9i)T + 3.48e3iT^{2} \)
61 \( 1 + (-39.2 - 39.2i)T + 3.72e3iT^{2} \)
67 \( 1 + (-68.6 - 68.6i)T + 4.48e3iT^{2} \)
71 \( 1 + 40.6T + 5.04e3T^{2} \)
73 \( 1 + 59.0iT - 5.32e3T^{2} \)
79 \( 1 + 17.3T + 6.24e3T^{2} \)
83 \( 1 + (-75.1 + 75.1i)T - 6.88e3iT^{2} \)
89 \( 1 - 78.8T + 7.92e3T^{2} \)
97 \( 1 + 38.8T + 9.40e3T^{2} \)
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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

−12.51274798771272904676175606371, −11.42461430594443546018159618636, −10.29411976675848176471144865381, −10.05197942352361360281944861376, −8.981102829958163908851309317914, −7.29490887946313932298775813869, −6.09911554270847758220018262699, −5.40473440215441547316529739911, −3.71255868839908112490763650475, −2.34632110917814461012773088662, 0.947615677416685273639975781063, 2.26395594157729484041766228028, 4.68777704979338337160627900663, 5.67832602391305649720587915049, 6.63539223168026691223367692186, 7.899052966255246042914415121524, 9.087397629123188104582817602878, 9.771656937236842272868147856964, 11.27693858541292827261598150011, 12.15509820382779710369768253624

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