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Properties

Label 2-192-192.59-c1-0-24
Degree $2$
Conductor $192$
Sign $0.842 + 0.538i$
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  + (1.19 − 0.752i)2-s + (0.486 + 1.66i)3-s + (0.866 − 1.80i)4-s + (0.594 − 2.98i)5-s + (1.83 + 1.62i)6-s + (−0.641 − 0.265i)7-s + (−0.319 − 2.81i)8-s + (−2.52 + 1.61i)9-s + (−1.53 − 4.02i)10-s + (3.20 + 4.79i)11-s + (3.41 + 0.564i)12-s + (−5.22 + 1.03i)13-s + (−0.968 + 0.164i)14-s + (5.25 − 0.464i)15-s + (−2.49 − 3.12i)16-s + (2.86 + 2.86i)17-s + ⋯
L(s)  = 1  + (0.846 − 0.532i)2-s + (0.280 + 0.959i)3-s + (0.433 − 0.901i)4-s + (0.265 − 1.33i)5-s + (0.748 + 0.663i)6-s + (−0.242 − 0.100i)7-s + (−0.112 − 0.993i)8-s + (−0.842 + 0.538i)9-s + (−0.486 − 1.27i)10-s + (0.965 + 1.44i)11-s + (0.986 + 0.162i)12-s + (−1.44 + 0.288i)13-s + (−0.258 + 0.0440i)14-s + (1.35 − 0.119i)15-s + (−0.624 − 0.781i)16-s + (0.694 + 0.694i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.842 + 0.538i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.842 + 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.842 + 0.538i$
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: \((2,\ 192,\ (\ :1/2),\ 0.842 + 0.538i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87427 - 0.548214i\)
\(L(\frac12)\) \(\approx\) \(1.87427 - 0.548214i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.19 + 0.752i)T \)
3 \( 1 + (-0.486 - 1.66i)T \)
good5 \( 1 + (-0.594 + 2.98i)T + (-4.61 - 1.91i)T^{2} \)
7 \( 1 + (0.641 + 0.265i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (-3.20 - 4.79i)T + (-4.20 + 10.1i)T^{2} \)
13 \( 1 + (5.22 - 1.03i)T + (12.0 - 4.97i)T^{2} \)
17 \( 1 + (-2.86 - 2.86i)T + 17iT^{2} \)
19 \( 1 + (-0.575 + 0.114i)T + (17.5 - 7.27i)T^{2} \)
23 \( 1 + (1.37 - 0.568i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (-0.996 - 0.666i)T + (11.0 + 26.7i)T^{2} \)
31 \( 1 + 5.17T + 31T^{2} \)
37 \( 1 + (0.334 - 1.68i)T + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (3.93 + 9.50i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (-1.48 - 2.22i)T + (-16.4 + 39.7i)T^{2} \)
47 \( 1 + (-5.25 + 5.25i)T - 47iT^{2} \)
53 \( 1 + (2.71 - 1.81i)T + (20.2 - 48.9i)T^{2} \)
59 \( 1 + (-12.3 - 2.45i)T + (54.5 + 22.5i)T^{2} \)
61 \( 1 + (-9.38 - 6.27i)T + (23.3 + 56.3i)T^{2} \)
67 \( 1 + (4.13 - 6.19i)T + (-25.6 - 61.8i)T^{2} \)
71 \( 1 + (-0.217 + 0.525i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (4.07 + 9.83i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-3.07 + 3.07i)T - 79iT^{2} \)
83 \( 1 + (0.885 + 4.45i)T + (-76.6 + 31.7i)T^{2} \)
89 \( 1 + (-1.45 + 3.51i)T + (-62.9 - 62.9i)T^{2} \)
97 \( 1 + 13.4iT - 97T^{2} \)
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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

−12.32604363759960926533414059740, −11.87028331503214887250472857722, −10.23627421049830769639245563173, −9.712412708745279889176693702882, −8.901510476923766658338547680682, −7.17959655372960500734424663671, −5.52044570470408017058847345127, −4.71584943219269156626557474476, −3.85808905419442632956080062120, −1.98217707309507571267657443572, 2.61939173120013102624062580189, 3.42559218638957833248692847449, 5.55966724436660299227704837654, 6.48756632797763019163679074021, 7.18258490446027227566069427900, 8.156770601669318053836116798446, 9.544053065653139088278075927022, 11.11005674512873921674112929438, 11.83394531653340552387737154758, 12.76088499063281924610549295749

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