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Properties

Label 2-192-64.29-c1-0-7
Degree $2$
Conductor $192$
Sign $-0.0849 + 0.996i$
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  + (−0.968 − 1.03i)2-s + (−0.980 − 0.195i)3-s + (−0.122 + 1.99i)4-s + (−0.0578 + 0.0866i)5-s + (0.749 + 1.19i)6-s + (2.40 − 0.997i)7-s + (2.17 − 1.80i)8-s + (0.923 + 0.382i)9-s + (0.145 − 0.0243i)10-s + (−1.19 − 5.98i)11-s + (0.509 − 1.93i)12-s + (−0.590 − 0.883i)13-s + (−3.36 − 1.51i)14-s + (0.0736 − 0.0736i)15-s + (−3.97 − 0.487i)16-s + (0.156 + 0.156i)17-s + ⋯
L(s)  = 1  + (−0.685 − 0.728i)2-s + (−0.566 − 0.112i)3-s + (−0.0610 + 0.998i)4-s + (−0.0258 + 0.0387i)5-s + (0.305 + 0.489i)6-s + (0.910 − 0.377i)7-s + (0.768 − 0.639i)8-s + (0.307 + 0.127i)9-s + (0.0459 − 0.00768i)10-s + (−0.359 − 1.80i)11-s + (0.146 − 0.558i)12-s + (−0.163 − 0.245i)13-s + (−0.898 − 0.404i)14-s + (0.0190 − 0.0190i)15-s + (−0.992 − 0.121i)16-s + (0.0379 + 0.0379i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0849 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0849 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-0.0849 + 0.996i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1/2),\ -0.0849 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.491824 - 0.535556i\)
\(L(\frac12)\) \(\approx\) \(0.491824 - 0.535556i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.968 + 1.03i)T \)
3 \( 1 + (0.980 + 0.195i)T \)
good5 \( 1 + (0.0578 - 0.0866i)T + (-1.91 - 4.61i)T^{2} \)
7 \( 1 + (-2.40 + 0.997i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (1.19 + 5.98i)T + (-10.1 + 4.20i)T^{2} \)
13 \( 1 + (0.590 + 0.883i)T + (-4.97 + 12.0i)T^{2} \)
17 \( 1 + (-0.156 - 0.156i)T + 17iT^{2} \)
19 \( 1 + (-3.11 + 2.08i)T + (7.27 - 17.5i)T^{2} \)
23 \( 1 + (-0.522 + 1.26i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-1.10 + 5.54i)T + (-26.7 - 11.0i)T^{2} \)
31 \( 1 + 7.51iT - 31T^{2} \)
37 \( 1 + (-6.15 - 4.11i)T + (14.1 + 34.1i)T^{2} \)
41 \( 1 + (1.10 - 2.66i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (6.90 - 1.37i)T + (39.7 - 16.4i)T^{2} \)
47 \( 1 + (-8.09 - 8.09i)T + 47iT^{2} \)
53 \( 1 + (-0.714 - 3.59i)T + (-48.9 + 20.2i)T^{2} \)
59 \( 1 + (6.55 - 9.80i)T + (-22.5 - 54.5i)T^{2} \)
61 \( 1 + (2.79 + 0.555i)T + (56.3 + 23.3i)T^{2} \)
67 \( 1 + (-0.861 - 0.171i)T + (61.8 + 25.6i)T^{2} \)
71 \( 1 + (7.68 - 3.18i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (5.79 + 2.39i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (-12.1 + 12.1i)T - 79iT^{2} \)
83 \( 1 + (7.21 - 4.82i)T + (31.7 - 76.6i)T^{2} \)
89 \( 1 + (-3.71 - 8.96i)T + (-62.9 + 62.9i)T^{2} \)
97 \( 1 - 4.45iT - 97T^{2} \)
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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

−11.82760697768985286988876470262, −11.22949481070676939451846474369, −10.63241184023283089010052896627, −9.398776602476101066906623256500, −8.216025196267714064155966766690, −7.51874076457194358556703202962, −5.94749460534592897195520764832, −4.52996697122977225047302466028, −2.96845087224810898852249835108, −0.956590151824249191691482786160, 1.80172804236876633223542538423, 4.67746200359123720679424380151, 5.33274991430991760805362716757, 6.80488160742756362611101899743, 7.60051552368642261558123092056, 8.738380940486835369533725815510, 9.851799877421145206130637521122, 10.57898620660380488688938780482, 11.74784025761406094051057159239, 12.57938403807233805169894255760

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