Properties

Label 2-192-192.107-c1-0-22
Degree $2$
Conductor $192$
Sign $-0.148 + 0.988i$
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  + (1.13 − 0.845i)2-s + (−1.46 − 0.917i)3-s + (0.569 − 1.91i)4-s + (0.908 + 1.35i)5-s + (−2.44 + 0.201i)6-s + (1.44 − 3.49i)7-s + (−0.975 − 2.65i)8-s + (1.31 + 2.69i)9-s + (2.17 + 0.772i)10-s + (−3.39 − 0.675i)11-s + (−2.59 + 2.29i)12-s + (−1.72 − 1.15i)13-s + (−1.31 − 5.18i)14-s + (−0.0863 − 2.82i)15-s + (−3.35 − 2.18i)16-s + (1.04 + 1.04i)17-s + ⋯
L(s)  = 1  + (0.801 − 0.597i)2-s + (−0.848 − 0.529i)3-s + (0.284 − 0.958i)4-s + (0.406 + 0.607i)5-s + (−0.996 + 0.0822i)6-s + (0.547 − 1.32i)7-s + (−0.344 − 0.938i)8-s + (0.438 + 0.898i)9-s + (0.688 + 0.244i)10-s + (−1.02 − 0.203i)11-s + (−0.749 + 0.661i)12-s + (−0.477 − 0.319i)13-s + (−0.351 − 1.38i)14-s + (−0.0222 − 0.730i)15-s + (−0.837 − 0.546i)16-s + (0.254 + 0.254i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.148 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.148 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-0.148 + 0.988i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1/2),\ -0.148 + 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.965425 - 1.12138i\)
\(L(\frac12)\) \(\approx\) \(0.965425 - 1.12138i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.13 + 0.845i)T \)
3 \( 1 + (1.46 + 0.917i)T \)
good5 \( 1 + (-0.908 - 1.35i)T + (-1.91 + 4.61i)T^{2} \)
7 \( 1 + (-1.44 + 3.49i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (3.39 + 0.675i)T + (10.1 + 4.20i)T^{2} \)
13 \( 1 + (1.72 + 1.15i)T + (4.97 + 12.0i)T^{2} \)
17 \( 1 + (-1.04 - 1.04i)T + 17iT^{2} \)
19 \( 1 + (-6.92 - 4.62i)T + (7.27 + 17.5i)T^{2} \)
23 \( 1 + (-1.88 - 4.54i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (-1.76 - 8.89i)T + (-26.7 + 11.0i)T^{2} \)
31 \( 1 - 2.37T + 31T^{2} \)
37 \( 1 + (-1.65 - 2.47i)T + (-14.1 + 34.1i)T^{2} \)
41 \( 1 + (-1.72 + 0.713i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (9.26 + 1.84i)T + (39.7 + 16.4i)T^{2} \)
47 \( 1 + (0.0451 - 0.0451i)T - 47iT^{2} \)
53 \( 1 + (-1.12 + 5.67i)T + (-48.9 - 20.2i)T^{2} \)
59 \( 1 + (5.63 - 3.76i)T + (22.5 - 54.5i)T^{2} \)
61 \( 1 + (-1.08 - 5.44i)T + (-56.3 + 23.3i)T^{2} \)
67 \( 1 + (8.44 - 1.67i)T + (61.8 - 25.6i)T^{2} \)
71 \( 1 + (7.65 + 3.16i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (-6.47 + 2.68i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (4.09 - 4.09i)T - 79iT^{2} \)
83 \( 1 + (-3.39 + 5.07i)T + (-31.7 - 76.6i)T^{2} \)
89 \( 1 + (2.79 + 1.15i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 + 5.24iT - 97T^{2} \)
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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

−12.23962673175445008592857339789, −11.33556870351336754272359494092, −10.37600777194497827140052261459, −10.15674431624026908456436165398, −7.72831607189940292442212175716, −6.97505218396472192640614199223, −5.67523793582557770054559514130, −4.85325432647525835555800498018, −3.18451307629382484344693390047, −1.35572446044170494206224679652, 2.70332771322452906523422384065, 4.82270784577291699548531710030, 5.13468028257610572476725286018, 6.14135657766646522089303557248, 7.52995392719229922757932182310, 8.803856203254561130091451508270, 9.737139858756364787851398017113, 11.28180301798989764511638595711, 11.93756605573543581788737760932, 12.70948856685989585555314184742

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