Properties

Label 2-192-64.29-c1-0-10
Degree $2$
Conductor $192$
Sign $0.951 - 0.307i$
Analytic cond. $1.53312$
Root an. cond. $1.23819$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.662 + 1.24i)2-s + (0.980 + 0.195i)3-s + (−1.12 − 1.65i)4-s + (1.62 − 2.43i)5-s + (−0.893 + 1.09i)6-s + (0.294 − 0.121i)7-s + (2.81 − 0.303i)8-s + (0.923 + 0.382i)9-s + (1.96 + 3.64i)10-s + (−1.07 − 5.42i)11-s + (−0.776 − 1.84i)12-s + (2.67 + 4.00i)13-s + (−0.0427 + 0.448i)14-s + (2.06 − 2.06i)15-s + (−1.48 + 3.71i)16-s + (0.394 + 0.394i)17-s + ⋯
L(s)  = 1  + (−0.468 + 0.883i)2-s + (0.566 + 0.112i)3-s + (−0.560 − 0.828i)4-s + (0.726 − 1.08i)5-s + (−0.364 + 0.447i)6-s + (0.111 − 0.0460i)7-s + (0.994 − 0.107i)8-s + (0.307 + 0.127i)9-s + (0.620 + 1.15i)10-s + (−0.325 − 1.63i)11-s + (−0.224 − 0.532i)12-s + (0.742 + 1.11i)13-s + (−0.0114 + 0.119i)14-s + (0.534 − 0.534i)15-s + (−0.371 + 0.928i)16-s + (0.0956 + 0.0956i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.307i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 - 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.951 - 0.307i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1/2),\ 0.951 - 0.307i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17854 + 0.185972i\)
\(L(\frac12)\) \(\approx\) \(1.17854 + 0.185972i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.662 - 1.24i)T \)
3 \( 1 + (-0.980 - 0.195i)T \)
good5 \( 1 + (-1.62 + 2.43i)T + (-1.91 - 4.61i)T^{2} \)
7 \( 1 + (-0.294 + 0.121i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (1.07 + 5.42i)T + (-10.1 + 4.20i)T^{2} \)
13 \( 1 + (-2.67 - 4.00i)T + (-4.97 + 12.0i)T^{2} \)
17 \( 1 + (-0.394 - 0.394i)T + 17iT^{2} \)
19 \( 1 + (2.13 - 1.42i)T + (7.27 - 17.5i)T^{2} \)
23 \( 1 + (3.37 - 8.14i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-0.411 + 2.06i)T + (-26.7 - 11.0i)T^{2} \)
31 \( 1 - 1.31iT - 31T^{2} \)
37 \( 1 + (-1.47 - 0.985i)T + (14.1 + 34.1i)T^{2} \)
41 \( 1 + (-4.39 + 10.6i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (2.25 - 0.448i)T + (39.7 - 16.4i)T^{2} \)
47 \( 1 + (5.23 + 5.23i)T + 47iT^{2} \)
53 \( 1 + (-1.88 - 9.48i)T + (-48.9 + 20.2i)T^{2} \)
59 \( 1 + (0.373 - 0.559i)T + (-22.5 - 54.5i)T^{2} \)
61 \( 1 + (-0.868 - 0.172i)T + (56.3 + 23.3i)T^{2} \)
67 \( 1 + (-10.3 - 2.04i)T + (61.8 + 25.6i)T^{2} \)
71 \( 1 + (12.4 - 5.14i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (14.5 + 6.01i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (2.55 - 2.55i)T - 79iT^{2} \)
83 \( 1 + (-0.585 + 0.391i)T + (31.7 - 76.6i)T^{2} \)
89 \( 1 + (-3.72 - 8.98i)T + (-62.9 + 62.9i)T^{2} \)
97 \( 1 + 0.565iT - 97T^{2} \)
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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

−13.11190636734431762939771475807, −11.46675312856691627728944358607, −10.26401879317275472515813093156, −9.142831042440713037950059770748, −8.723598451320258031736043241410, −7.77016604930054822889678573253, −6.19883870616023039094781469260, −5.42032134508362379393035155445, −3.98192082552419725095957937372, −1.50169079904368332106909589720, 2.06731955118537593672009647082, 3.01407913868063894585617439972, 4.58375552892895446224918453452, 6.46362190433189498440202629047, 7.61289739253721692827886320625, 8.574187865129933262855337634128, 9.973384078191791899665211146471, 10.20551808130957762823837857789, 11.28793307679865917144465427419, 12.69495690650349473052460996406

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