Properties

Label 2-192-192.35-c1-0-1
Degree $2$
Conductor $192$
Sign $-0.00990 - 0.999i$
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  + (−1.35 − 0.411i)2-s + (−0.145 + 1.72i)3-s + (1.66 + 1.11i)4-s + (0.731 + 0.488i)5-s + (0.907 − 2.27i)6-s + (0.683 + 1.64i)7-s + (−1.78 − 2.18i)8-s + (−2.95 − 0.503i)9-s + (−0.788 − 0.962i)10-s + (0.385 + 1.93i)11-s + (−2.16 + 2.70i)12-s + (−0.659 − 0.987i)13-s + (−0.245 − 2.51i)14-s + (−0.950 + 1.19i)15-s + (1.52 + 3.69i)16-s + (−2.96 + 2.96i)17-s + ⋯
L(s)  = 1  + (−0.956 − 0.290i)2-s + (−0.0841 + 0.996i)3-s + (0.830 + 0.556i)4-s + (0.327 + 0.218i)5-s + (0.370 − 0.928i)6-s + (0.258 + 0.623i)7-s + (−0.632 − 0.774i)8-s + (−0.985 − 0.167i)9-s + (−0.249 − 0.304i)10-s + (0.116 + 0.584i)11-s + (−0.624 + 0.780i)12-s + (−0.182 − 0.273i)13-s + (−0.0656 − 0.671i)14-s + (−0.245 + 0.307i)15-s + (0.380 + 0.924i)16-s + (−0.720 + 0.720i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00990 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00990 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-0.00990 - 0.999i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1/2),\ -0.00990 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.527105 + 0.532354i\)
\(L(\frac12)\) \(\approx\) \(0.527105 + 0.532354i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.35 + 0.411i)T \)
3 \( 1 + (0.145 - 1.72i)T \)
good5 \( 1 + (-0.731 - 0.488i)T + (1.91 + 4.61i)T^{2} \)
7 \( 1 + (-0.683 - 1.64i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (-0.385 - 1.93i)T + (-10.1 + 4.20i)T^{2} \)
13 \( 1 + (0.659 + 0.987i)T + (-4.97 + 12.0i)T^{2} \)
17 \( 1 + (2.96 - 2.96i)T - 17iT^{2} \)
19 \( 1 + (-2.88 - 4.31i)T + (-7.27 + 17.5i)T^{2} \)
23 \( 1 + (1.53 - 3.71i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-8.74 - 1.74i)T + (26.7 + 11.0i)T^{2} \)
31 \( 1 - 2.66T + 31T^{2} \)
37 \( 1 + (2.69 + 1.80i)T + (14.1 + 34.1i)T^{2} \)
41 \( 1 + (9.64 + 3.99i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (0.964 + 4.84i)T + (-39.7 + 16.4i)T^{2} \)
47 \( 1 + (-2.39 - 2.39i)T + 47iT^{2} \)
53 \( 1 + (-13.1 + 2.61i)T + (48.9 - 20.2i)T^{2} \)
59 \( 1 + (-5.24 + 7.85i)T + (-22.5 - 54.5i)T^{2} \)
61 \( 1 + (-4.01 - 0.798i)T + (56.3 + 23.3i)T^{2} \)
67 \( 1 + (1.21 - 6.11i)T + (-61.8 - 25.6i)T^{2} \)
71 \( 1 + (-8.97 + 3.71i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (3.96 + 1.64i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (-3.91 - 3.91i)T + 79iT^{2} \)
83 \( 1 + (-7.85 + 5.24i)T + (31.7 - 76.6i)T^{2} \)
89 \( 1 + (-7.76 + 3.21i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 - 2.84iT - 97T^{2} \)
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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

−12.23044961677075711033815575761, −11.72054162435155372574900886891, −10.36250485026216348610000606695, −10.07583382022994134721741700701, −8.903514204960191330891536905423, −8.141955752395474274628492806017, −6.61845048263880329221778122232, −5.40633079638498230907435984635, −3.76030694852704695993442812674, −2.25702510744132245464946040142, 0.942115640172179462051070332463, 2.59064323372618134419543439552, 5.10053257828298137005958034847, 6.45568833999141323872844461976, 7.13868832572746729980146004821, 8.247224648920388336435316999859, 9.054773679771665947045467504675, 10.27429753968105471712410105563, 11.35720724602707027398446161800, 11.98150230677036951032497358192

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