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Properties

Label 2-192-192.155-c1-0-20
Degree $2$
Conductor $192$
Sign $0.908 - 0.417i$
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  + (1.11 + 0.875i)2-s + (1.20 − 1.24i)3-s + (0.466 + 1.94i)4-s + (1.81 + 0.361i)5-s + (2.42 − 0.334i)6-s + (−2.53 − 1.04i)7-s + (−1.18 + 2.56i)8-s + (−0.114 − 2.99i)9-s + (1.70 + 1.99i)10-s + (−4.03 + 2.69i)11-s + (2.98 + 1.75i)12-s + (−1.10 − 5.57i)13-s + (−1.89 − 3.38i)14-s + (2.63 − 1.83i)15-s + (−3.56 + 1.81i)16-s + (3.00 + 3.00i)17-s + ⋯
L(s)  = 1  + (0.785 + 0.619i)2-s + (0.693 − 0.720i)3-s + (0.233 + 0.972i)4-s + (0.813 + 0.161i)5-s + (0.990 − 0.136i)6-s + (−0.957 − 0.396i)7-s + (−0.418 + 0.908i)8-s + (−0.0381 − 0.999i)9-s + (0.538 + 0.630i)10-s + (−1.21 + 0.813i)11-s + (0.862 + 0.506i)12-s + (−0.307 − 1.54i)13-s + (−0.506 − 0.904i)14-s + (0.680 − 0.473i)15-s + (−0.891 + 0.453i)16-s + (0.729 + 0.729i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 - 0.417i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.908 - 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.908 - 0.417i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1/2),\ 0.908 - 0.417i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.02289 + 0.442737i\)
\(L(\frac12)\) \(\approx\) \(2.02289 + 0.442737i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.11 - 0.875i)T \)
3 \( 1 + (-1.20 + 1.24i)T \)
good5 \( 1 + (-1.81 - 0.361i)T + (4.61 + 1.91i)T^{2} \)
7 \( 1 + (2.53 + 1.04i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (4.03 - 2.69i)T + (4.20 - 10.1i)T^{2} \)
13 \( 1 + (1.10 + 5.57i)T + (-12.0 + 4.97i)T^{2} \)
17 \( 1 + (-3.00 - 3.00i)T + 17iT^{2} \)
19 \( 1 + (-1.06 - 5.35i)T + (-17.5 + 7.27i)T^{2} \)
23 \( 1 + (-4.06 + 1.68i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (-2.31 + 3.46i)T + (-11.0 - 26.7i)T^{2} \)
31 \( 1 + 6.64T + 31T^{2} \)
37 \( 1 + (-1.21 - 0.241i)T + (34.1 + 14.1i)T^{2} \)
41 \( 1 + (-2.36 - 5.70i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (-5.44 + 3.64i)T + (16.4 - 39.7i)T^{2} \)
47 \( 1 + (1.67 - 1.67i)T - 47iT^{2} \)
53 \( 1 + (-1.54 - 2.31i)T + (-20.2 + 48.9i)T^{2} \)
59 \( 1 + (-1.24 + 6.25i)T + (-54.5 - 22.5i)T^{2} \)
61 \( 1 + (-1.32 + 1.98i)T + (-23.3 - 56.3i)T^{2} \)
67 \( 1 + (2.24 + 1.50i)T + (25.6 + 61.8i)T^{2} \)
71 \( 1 + (1.46 - 3.52i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-4.57 - 11.0i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-3.67 + 3.67i)T - 79iT^{2} \)
83 \( 1 + (-15.2 + 3.03i)T + (76.6 - 31.7i)T^{2} \)
89 \( 1 + (1.41 - 3.41i)T + (-62.9 - 62.9i)T^{2} \)
97 \( 1 - 2.24iT - 97T^{2} \)
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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

−12.83174399364424912364896229997, −12.40953646827885578472489614284, −10.44745644499415868606882826611, −9.657339985078604970994936169865, −8.059738960941354172007551500732, −7.50978527149942376445801583165, −6.29819155475099087662156448232, −5.44474987323514763783295651041, −3.53618033278033196751822065334, −2.49034747428571713475594001645, 2.39435449596499462322589663463, 3.30246799266386807085274816695, 4.88440698514312834751776490263, 5.71209841645939698538761335125, 7.15269392923175727551349662806, 9.201352127480836795073860093251, 9.359674454286605014270993308219, 10.51234015233963161856230711467, 11.41813868363149816909619831141, 12.75154183678558249280054115277

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