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Properties

Label 2-192-192.107-c1-0-1
Degree $2$
Conductor $192$
Sign $0.867 - 0.497i$
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  + (−0.743 − 1.20i)2-s + (−1.02 − 1.39i)3-s + (−0.894 + 1.78i)4-s + (1.69 + 2.52i)5-s + (−0.911 + 2.27i)6-s + (−1.66 + 4.02i)7-s + (2.81 − 0.253i)8-s + (−0.884 + 2.86i)9-s + (1.78 − 3.91i)10-s + (−3.48 − 0.692i)11-s + (3.41 − 0.593i)12-s + (3.08 + 2.06i)13-s + (6.08 − 0.987i)14-s + (1.78 − 4.95i)15-s + (−2.39 − 3.20i)16-s + (−2.62 − 2.62i)17-s + ⋯
L(s)  = 1  + (−0.525 − 0.850i)2-s + (−0.593 − 0.804i)3-s + (−0.447 + 0.894i)4-s + (0.755 + 1.13i)5-s + (−0.372 + 0.928i)6-s + (−0.630 + 1.52i)7-s + (0.995 − 0.0896i)8-s + (−0.294 + 0.955i)9-s + (0.565 − 1.23i)10-s + (−1.04 − 0.208i)11-s + (0.985 − 0.171i)12-s + (0.856 + 0.572i)13-s + (1.62 − 0.264i)14-s + (0.461 − 1.28i)15-s + (−0.599 − 0.800i)16-s + (−0.636 − 0.636i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 - 0.497i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.867 - 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.867 - 0.497i$
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: \((2,\ 192,\ (\ :1/2),\ 0.867 - 0.497i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.627307 + 0.167177i\)
\(L(\frac12)\) \(\approx\) \(0.627307 + 0.167177i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.743 + 1.20i)T \)
3 \( 1 + (1.02 + 1.39i)T \)
good5 \( 1 + (-1.69 - 2.52i)T + (-1.91 + 4.61i)T^{2} \)
7 \( 1 + (1.66 - 4.02i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (3.48 + 0.692i)T + (10.1 + 4.20i)T^{2} \)
13 \( 1 + (-3.08 - 2.06i)T + (4.97 + 12.0i)T^{2} \)
17 \( 1 + (2.62 + 2.62i)T + 17iT^{2} \)
19 \( 1 + (-2.52 - 1.68i)T + (7.27 + 17.5i)T^{2} \)
23 \( 1 + (-1.33 - 3.23i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (0.240 + 1.20i)T + (-26.7 + 11.0i)T^{2} \)
31 \( 1 + 5.17T + 31T^{2} \)
37 \( 1 + (-2.19 - 3.29i)T + (-14.1 + 34.1i)T^{2} \)
41 \( 1 + (2.03 - 0.842i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (-10.1 - 2.01i)T + (39.7 + 16.4i)T^{2} \)
47 \( 1 + (-3.16 + 3.16i)T - 47iT^{2} \)
53 \( 1 + (1.90 - 9.57i)T + (-48.9 - 20.2i)T^{2} \)
59 \( 1 + (-8.20 + 5.48i)T + (22.5 - 54.5i)T^{2} \)
61 \( 1 + (1.36 + 6.84i)T + (-56.3 + 23.3i)T^{2} \)
67 \( 1 + (-3.78 + 0.753i)T + (61.8 - 25.6i)T^{2} \)
71 \( 1 + (4.21 + 1.74i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (1.11 - 0.460i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (-2.41 + 2.41i)T - 79iT^{2} \)
83 \( 1 + (2.81 - 4.21i)T + (-31.7 - 76.6i)T^{2} \)
89 \( 1 + (-6.04 - 2.50i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 - 3.52iT - 97T^{2} \)
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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

−12.49639567248276201636271477465, −11.47090577286521253011404811656, −10.89451650393986723862941401384, −9.766066863077888174769953836530, −8.821087610452111802552074607619, −7.52540962339633717043975973347, −6.37234218282630134827618549084, −5.43528126589935102036743485124, −2.97950600735445273846246437102, −2.13316155677345113257032073023, 0.74227310090101111145301788501, 4.08522759419009646017213548539, 5.15961763259285910273983246999, 6.03897282664147716908698753558, 7.25754730721404914074573025873, 8.616245967195688578130244515916, 9.477466835448543499592445454678, 10.40245326695980513400987719804, 10.84052769712153225543565543176, 12.96500208565349974266525614241

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