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Properties

Label 2-192-192.59-c1-0-16
Degree $2$
Conductor $192$
Sign $0.430 + 0.902i$
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  + (−1.07 − 0.914i)2-s + (1.48 + 0.889i)3-s + (0.327 + 1.97i)4-s + (0.737 − 3.70i)5-s + (−0.789 − 2.31i)6-s + (−2.11 − 0.877i)7-s + (1.45 − 2.42i)8-s + (1.41 + 2.64i)9-s + (−4.18 + 3.32i)10-s + (−1.12 − 1.68i)11-s + (−1.26 + 3.22i)12-s + (6.48 − 1.28i)13-s + (1.48 + 2.88i)14-s + (4.39 − 4.85i)15-s + (−3.78 + 1.29i)16-s + (2.20 + 2.20i)17-s + ⋯
L(s)  = 1  + (−0.762 − 0.646i)2-s + (0.857 + 0.513i)3-s + (0.163 + 0.986i)4-s + (0.329 − 1.65i)5-s + (−0.322 − 0.946i)6-s + (−0.800 − 0.331i)7-s + (0.512 − 0.858i)8-s + (0.472 + 0.881i)9-s + (−1.32 + 1.05i)10-s + (−0.338 − 0.506i)11-s + (−0.366 + 0.930i)12-s + (1.79 − 0.357i)13-s + (0.396 + 0.770i)14-s + (1.13 − 1.25i)15-s + (−0.946 + 0.323i)16-s + (0.534 + 0.534i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.430 + 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.430 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.430 + 0.902i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1/2),\ 0.430 + 0.902i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.917164 - 0.578430i\)
\(L(\frac12)\) \(\approx\) \(0.917164 - 0.578430i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.07 + 0.914i)T \)
3 \( 1 + (-1.48 - 0.889i)T \)
good5 \( 1 + (-0.737 + 3.70i)T + (-4.61 - 1.91i)T^{2} \)
7 \( 1 + (2.11 + 0.877i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (1.12 + 1.68i)T + (-4.20 + 10.1i)T^{2} \)
13 \( 1 + (-6.48 + 1.28i)T + (12.0 - 4.97i)T^{2} \)
17 \( 1 + (-2.20 - 2.20i)T + 17iT^{2} \)
19 \( 1 + (1.20 - 0.238i)T + (17.5 - 7.27i)T^{2} \)
23 \( 1 + (3.08 - 1.27i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (2.30 + 1.54i)T + (11.0 + 26.7i)T^{2} \)
31 \( 1 - 3.90T + 31T^{2} \)
37 \( 1 + (0.767 - 3.86i)T + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (-0.850 - 2.05i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (-3.77 - 5.64i)T + (-16.4 + 39.7i)T^{2} \)
47 \( 1 + (5.22 - 5.22i)T - 47iT^{2} \)
53 \( 1 + (8.91 - 5.95i)T + (20.2 - 48.9i)T^{2} \)
59 \( 1 + (-0.934 - 0.185i)T + (54.5 + 22.5i)T^{2} \)
61 \( 1 + (-6.48 - 4.33i)T + (23.3 + 56.3i)T^{2} \)
67 \( 1 + (3.71 - 5.56i)T + (-25.6 - 61.8i)T^{2} \)
71 \( 1 + (-4.33 + 10.4i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (1.32 + 3.19i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (7.64 - 7.64i)T - 79iT^{2} \)
83 \( 1 + (-1.05 - 5.28i)T + (-76.6 + 31.7i)T^{2} \)
89 \( 1 + (2.02 - 4.89i)T + (-62.9 - 62.9i)T^{2} \)
97 \( 1 - 4.05iT - 97T^{2} \)
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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

−12.59323570000078926941326669077, −11.14025834063996516658111638689, −10.08669398768964674623911020001, −9.392797634318490488366960244103, −8.441938996354164817696143006432, −8.043422388127202009090746694223, −6.01008291096148304947200664799, −4.30774681117490688061612923096, −3.28562715260794491155461352667, −1.34696648931508498413434755461, 2.14809170205590902110627533495, 3.45310556167114290519513292365, 6.04579801556695415947879380340, 6.64507340736650310665872654039, 7.51640957294817704430779698642, 8.643905061825060447546196597504, 9.665083232796588897514999042149, 10.40113120055492771974830182886, 11.47817360099143551670106144395, 13.03403229219069498042294132646

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