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Properties

Label 2-192-192.131-c1-0-15
Degree $2$
Conductor $192$
Sign $0.965 - 0.261i$
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  + (−1.04 + 0.954i)2-s + (1.68 − 0.409i)3-s + (0.179 − 1.99i)4-s + (0.0417 − 0.0624i)5-s + (−1.36 + 2.03i)6-s + (−0.100 − 0.242i)7-s + (1.71 + 2.25i)8-s + (2.66 − 1.37i)9-s + (0.0160 + 0.104i)10-s + (2.15 − 0.428i)11-s + (−0.514 − 3.42i)12-s + (0.751 − 0.502i)13-s + (0.335 + 0.157i)14-s + (0.0445 − 0.122i)15-s + (−3.93 − 0.714i)16-s + (1.66 − 1.66i)17-s + ⋯
L(s)  = 1  + (−0.738 + 0.674i)2-s + (0.971 − 0.236i)3-s + (0.0896 − 0.995i)4-s + (0.0186 − 0.0279i)5-s + (−0.557 + 0.830i)6-s + (−0.0379 − 0.0915i)7-s + (0.605 + 0.795i)8-s + (0.887 − 0.459i)9-s + (0.00506 + 0.0331i)10-s + (0.650 − 0.129i)11-s + (−0.148 − 0.988i)12-s + (0.208 − 0.139i)13-s + (0.0897 + 0.0419i)14-s + (0.0115 − 0.0315i)15-s + (−0.983 − 0.178i)16-s + (0.404 − 0.404i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.261i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.965 - 0.261i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1/2),\ 0.965 - 0.261i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15226 + 0.153112i\)
\(L(\frac12)\) \(\approx\) \(1.15226 + 0.153112i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.04 - 0.954i)T \)
3 \( 1 + (-1.68 + 0.409i)T \)
good5 \( 1 + (-0.0417 + 0.0624i)T + (-1.91 - 4.61i)T^{2} \)
7 \( 1 + (0.100 + 0.242i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (-2.15 + 0.428i)T + (10.1 - 4.20i)T^{2} \)
13 \( 1 + (-0.751 + 0.502i)T + (4.97 - 12.0i)T^{2} \)
17 \( 1 + (-1.66 + 1.66i)T - 17iT^{2} \)
19 \( 1 + (1.06 - 0.711i)T + (7.27 - 17.5i)T^{2} \)
23 \( 1 + (0.745 - 1.79i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (1.26 - 6.37i)T + (-26.7 - 11.0i)T^{2} \)
31 \( 1 + 10.1T + 31T^{2} \)
37 \( 1 + (-2.54 + 3.81i)T + (-14.1 - 34.1i)T^{2} \)
41 \( 1 + (4.77 + 1.97i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (9.37 - 1.86i)T + (39.7 - 16.4i)T^{2} \)
47 \( 1 + (8.04 + 8.04i)T + 47iT^{2} \)
53 \( 1 + (-1.29 - 6.50i)T + (-48.9 + 20.2i)T^{2} \)
59 \( 1 + (3.86 + 2.57i)T + (22.5 + 54.5i)T^{2} \)
61 \( 1 + (0.853 - 4.29i)T + (-56.3 - 23.3i)T^{2} \)
67 \( 1 + (-1.79 - 0.357i)T + (61.8 + 25.6i)T^{2} \)
71 \( 1 + (5.21 - 2.16i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (-10.0 - 4.15i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (5.93 + 5.93i)T + 79iT^{2} \)
83 \( 1 + (-6.79 - 10.1i)T + (-31.7 + 76.6i)T^{2} \)
89 \( 1 + (13.5 - 5.62i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 + 13.0iT - 97T^{2} \)
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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

−12.81559992549880492398851210117, −11.43249178052264947079566640150, −10.26185410562211587514620648874, −9.272957513801456853276113548274, −8.639805848477598489559862794567, −7.50790656075664310207289433918, −6.76575364936202117968089081709, −5.32570147398171134951117697235, −3.56848990707789563295173894504, −1.62416270592390354792297742996, 1.86443759989596285120920605918, 3.29958680240239825956658303356, 4.40483822240316065315564609760, 6.59335660891228562639466420748, 7.82006057617615135460680842334, 8.634707107461949315081008874861, 9.519667408241918879362689756163, 10.30086999956222926212618997087, 11.38119786157224084560885649432, 12.47176118773849618957239242770

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