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Properties

Label 2-192-64.37-c1-0-5
Degree $2$
Conductor $192$
Sign $0.764 - 0.644i$
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  + (1.40 − 0.187i)2-s + (−0.555 + 0.831i)3-s + (1.92 − 0.525i)4-s + (−0.451 + 2.26i)5-s + (−0.622 + 1.26i)6-s + (−1.32 + 3.19i)7-s + (2.60 − 1.09i)8-s + (−0.382 − 0.923i)9-s + (−0.207 + 3.26i)10-s + (3.07 − 2.05i)11-s + (−0.634 + 1.89i)12-s + (−1.14 − 5.77i)13-s + (−1.25 + 4.72i)14-s + (−1.63 − 1.63i)15-s + (3.44 − 2.02i)16-s + (3.36 − 3.36i)17-s + ⋯
L(s)  = 1  + (0.991 − 0.132i)2-s + (−0.320 + 0.480i)3-s + (0.964 − 0.262i)4-s + (−0.201 + 1.01i)5-s + (−0.254 + 0.518i)6-s + (−0.499 + 1.20i)7-s + (0.921 − 0.388i)8-s + (−0.127 − 0.307i)9-s + (−0.0654 + 1.03i)10-s + (0.926 − 0.618i)11-s + (−0.183 + 0.547i)12-s + (−0.318 − 1.60i)13-s + (−0.335 + 1.26i)14-s + (−0.422 − 0.422i)15-s + (0.861 − 0.507i)16-s + (0.816 − 0.816i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.764 - 0.644i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1/2),\ 0.764 - 0.644i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70624 + 0.622825i\)
\(L(\frac12)\) \(\approx\) \(1.70624 + 0.622825i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 + 0.187i)T \)
3 \( 1 + (0.555 - 0.831i)T \)
good5 \( 1 + (0.451 - 2.26i)T + (-4.61 - 1.91i)T^{2} \)
7 \( 1 + (1.32 - 3.19i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (-3.07 + 2.05i)T + (4.20 - 10.1i)T^{2} \)
13 \( 1 + (1.14 + 5.77i)T + (-12.0 + 4.97i)T^{2} \)
17 \( 1 + (-3.36 + 3.36i)T - 17iT^{2} \)
19 \( 1 + (6.10 - 1.21i)T + (17.5 - 7.27i)T^{2} \)
23 \( 1 + (1.30 - 0.539i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (0.714 + 0.477i)T + (11.0 + 26.7i)T^{2} \)
31 \( 1 - 7.07iT - 31T^{2} \)
37 \( 1 + (2.72 + 0.542i)T + (34.1 + 14.1i)T^{2} \)
41 \( 1 + (-1.11 + 0.462i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (6.65 + 9.95i)T + (-16.4 + 39.7i)T^{2} \)
47 \( 1 + (-2.55 + 2.55i)T - 47iT^{2} \)
53 \( 1 + (-6.68 + 4.46i)T + (20.2 - 48.9i)T^{2} \)
59 \( 1 + (1.36 - 6.86i)T + (-54.5 - 22.5i)T^{2} \)
61 \( 1 + (7.61 - 11.4i)T + (-23.3 - 56.3i)T^{2} \)
67 \( 1 + (1.53 - 2.29i)T + (-25.6 - 61.8i)T^{2} \)
71 \( 1 + (1.95 - 4.72i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (3.79 + 9.16i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-3.88 - 3.88i)T + 79iT^{2} \)
83 \( 1 + (-9.26 + 1.84i)T + (76.6 - 31.7i)T^{2} \)
89 \( 1 + (5.39 + 2.23i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 - 7.50iT - 97T^{2} \)
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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

−12.34554328254682557697319430612, −11.93344965323287419599469766374, −10.75252305579182580267831239529, −10.14694493537610980565507500474, −8.679291989567697557229850281789, −7.11838785074552243302128324785, −6.07398033382165369773012056479, −5.33343057812824406833579815223, −3.60171166983426308510256469640, −2.77742588469208957116516897583, 1.65750575510660002267218022571, 4.01845029967888817107384722545, 4.57965875701465148577154398684, 6.27238967537619118812571421900, 6.93333445011408837222234778434, 8.073561943596654710419355747920, 9.505706022655548251491295839987, 10.77927284200995870887600966172, 11.86152338236321736953172332954, 12.52352525699211598193129783483

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