Properties

Label 2-192-64.29-c1-0-13
Degree $2$
Conductor $192$
Sign $0.555 + 0.831i$
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  + (0.914 − 1.07i)2-s + (0.980 + 0.195i)3-s + (−0.325 − 1.97i)4-s + (−1.12 + 1.67i)5-s + (1.10 − 0.879i)6-s + (3.63 − 1.50i)7-s + (−2.42 − 1.45i)8-s + (0.923 + 0.382i)9-s + (0.783 + 2.74i)10-s + (0.181 + 0.913i)11-s + (0.0654 − 1.99i)12-s + (−2.94 − 4.40i)13-s + (1.70 − 5.29i)14-s + (−1.42 + 1.42i)15-s + (−3.78 + 1.28i)16-s + (2.67 + 2.67i)17-s + ⋯
L(s)  = 1  + (0.646 − 0.762i)2-s + (0.566 + 0.112i)3-s + (−0.162 − 0.986i)4-s + (−0.501 + 0.750i)5-s + (0.452 − 0.358i)6-s + (1.37 − 0.568i)7-s + (−0.857 − 0.514i)8-s + (0.307 + 0.127i)9-s + (0.247 + 0.867i)10-s + (0.0548 + 0.275i)11-s + (0.0188 − 0.577i)12-s + (−0.816 − 1.22i)13-s + (0.454 − 1.41i)14-s + (−0.368 + 0.368i)15-s + (−0.946 + 0.321i)16-s + (0.647 + 0.647i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.555 + 0.831i$
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: \((2,\ 192,\ (\ :1/2),\ 0.555 + 0.831i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61566 - 0.863600i\)
\(L(\frac12)\) \(\approx\) \(1.61566 - 0.863600i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.914 + 1.07i)T \)
3 \( 1 + (-0.980 - 0.195i)T \)
good5 \( 1 + (1.12 - 1.67i)T + (-1.91 - 4.61i)T^{2} \)
7 \( 1 + (-3.63 + 1.50i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (-0.181 - 0.913i)T + (-10.1 + 4.20i)T^{2} \)
13 \( 1 + (2.94 + 4.40i)T + (-4.97 + 12.0i)T^{2} \)
17 \( 1 + (-2.67 - 2.67i)T + 17iT^{2} \)
19 \( 1 + (6.32 - 4.22i)T + (7.27 - 17.5i)T^{2} \)
23 \( 1 + (3.18 - 7.69i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-0.769 + 3.86i)T + (-26.7 - 11.0i)T^{2} \)
31 \( 1 + 4.08iT - 31T^{2} \)
37 \( 1 + (-2.30 - 1.54i)T + (14.1 + 34.1i)T^{2} \)
41 \( 1 + (-0.846 + 2.04i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (-3.25 + 0.647i)T + (39.7 - 16.4i)T^{2} \)
47 \( 1 + (1.41 + 1.41i)T + 47iT^{2} \)
53 \( 1 + (2.22 + 11.1i)T + (-48.9 + 20.2i)T^{2} \)
59 \( 1 + (4.86 - 7.27i)T + (-22.5 - 54.5i)T^{2} \)
61 \( 1 + (7.71 + 1.53i)T + (56.3 + 23.3i)T^{2} \)
67 \( 1 + (-10.8 - 2.15i)T + (61.8 + 25.6i)T^{2} \)
71 \( 1 + (8.29 - 3.43i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (1.40 + 0.581i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (-7.55 + 7.55i)T - 79iT^{2} \)
83 \( 1 + (-3.83 + 2.56i)T + (31.7 - 76.6i)T^{2} \)
89 \( 1 + (2.90 + 7.01i)T + (-62.9 + 62.9i)T^{2} \)
97 \( 1 + 5.94iT - 97T^{2} \)
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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

−12.34520509094342812965642774733, −11.37505400016911591460110738012, −10.53179634832483546284077888940, −9.872910983680889185903720810725, −8.157020950839609029145760871878, −7.49004202768367172340437004625, −5.76732915102700888581277247625, −4.42744315082238295424344595261, −3.46442597693989909781215474088, −1.92021620124386544168703851796, 2.42657456451371677420339452810, 4.42685800476187952349813914739, 4.84476854274859383418376490847, 6.50972874191384263241051966681, 7.70740184129458314266730437975, 8.526720638005509941622774544361, 9.089576147155280763880666307735, 11.07044715122360769949189075127, 12.17031926550466242159992490809, 12.53955725867763938289152659832

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