Properties

Label 2-192-64.29-c1-0-11
Degree $2$
Conductor $192$
Sign $0.964 - 0.265i$
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  + (1.41 − 0.0401i)2-s + (0.980 + 0.195i)3-s + (1.99 − 0.113i)4-s + (−1.10 + 1.66i)5-s + (1.39 + 0.236i)6-s + (−1.97 + 0.816i)7-s + (2.81 − 0.240i)8-s + (0.923 + 0.382i)9-s + (−1.50 + 2.39i)10-s + (−0.842 − 4.23i)11-s + (1.98 + 0.278i)12-s + (0.0329 + 0.0493i)13-s + (−2.75 + 1.23i)14-s + (−1.41 + 1.41i)15-s + (3.97 − 0.453i)16-s + (−5.58 − 5.58i)17-s + ⋯
L(s)  = 1  + (0.999 − 0.0284i)2-s + (0.566 + 0.112i)3-s + (0.998 − 0.0568i)4-s + (−0.496 + 0.742i)5-s + (0.569 + 0.0964i)6-s + (−0.744 + 0.308i)7-s + (0.996 − 0.0851i)8-s + (0.307 + 0.127i)9-s + (−0.475 + 0.756i)10-s + (−0.254 − 1.27i)11-s + (0.571 + 0.0802i)12-s + (0.00914 + 0.0136i)13-s + (−0.735 + 0.329i)14-s + (−0.364 + 0.364i)15-s + (0.993 − 0.113i)16-s + (−1.35 − 1.35i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.964 - 0.265i$
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: \((2,\ 192,\ (\ :1/2),\ 0.964 - 0.265i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.12536 + 0.287280i\)
\(L(\frac12)\) \(\approx\) \(2.12536 + 0.287280i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 0.0401i)T \)
3 \( 1 + (-0.980 - 0.195i)T \)
good5 \( 1 + (1.10 - 1.66i)T + (-1.91 - 4.61i)T^{2} \)
7 \( 1 + (1.97 - 0.816i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (0.842 + 4.23i)T + (-10.1 + 4.20i)T^{2} \)
13 \( 1 + (-0.0329 - 0.0493i)T + (-4.97 + 12.0i)T^{2} \)
17 \( 1 + (5.58 + 5.58i)T + 17iT^{2} \)
19 \( 1 + (-0.497 + 0.332i)T + (7.27 - 17.5i)T^{2} \)
23 \( 1 + (-0.591 + 1.42i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (1.85 - 9.31i)T + (-26.7 - 11.0i)T^{2} \)
31 \( 1 + 2.71iT - 31T^{2} \)
37 \( 1 + (-7.23 - 4.83i)T + (14.1 + 34.1i)T^{2} \)
41 \( 1 + (3.74 - 9.02i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (-1.17 + 0.234i)T + (39.7 - 16.4i)T^{2} \)
47 \( 1 + (6.16 + 6.16i)T + 47iT^{2} \)
53 \( 1 + (0.912 + 4.58i)T + (-48.9 + 20.2i)T^{2} \)
59 \( 1 + (-1.50 + 2.25i)T + (-22.5 - 54.5i)T^{2} \)
61 \( 1 + (2.18 + 0.434i)T + (56.3 + 23.3i)T^{2} \)
67 \( 1 + (-14.1 - 2.80i)T + (61.8 + 25.6i)T^{2} \)
71 \( 1 + (0.535 - 0.221i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (2.44 + 1.01i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (6.21 - 6.21i)T - 79iT^{2} \)
83 \( 1 + (-4.16 + 2.78i)T + (31.7 - 76.6i)T^{2} \)
89 \( 1 + (0.479 + 1.15i)T + (-62.9 + 62.9i)T^{2} \)
97 \( 1 + 0.928iT - 97T^{2} \)
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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

−12.90987036411711229693834737609, −11.45060195364385447742672916683, −11.06105647549162812382992098523, −9.708116989416660986774247902570, −8.448897650411289370448216718951, −7.15764105875533061439496401147, −6.37916295131960944380204133996, −4.93280619326880792925984544036, −3.41805668256924962428202769128, −2.76715208608988357030418385402, 2.17163487405731760272271478642, 3.85585455308775781956475847127, 4.60145805069165273957185456833, 6.20480332260809578222820683549, 7.26976649902487843236694842379, 8.246821397080924675011273773442, 9.571970615210285962499715433024, 10.64384868045852832523154518316, 11.90527563668888275874567204212, 12.93559833268183857838145925904

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