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Properties

Label 2-192-192.107-c1-0-15
Degree $2$
Conductor $192$
Sign $0.480 - 0.876i$
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  + (1.33 + 0.454i)2-s + (0.322 + 1.70i)3-s + (1.58 + 1.21i)4-s + (−0.205 − 0.307i)5-s + (−0.342 + 2.42i)6-s + (1.38 − 3.34i)7-s + (1.56 + 2.35i)8-s + (−2.79 + 1.09i)9-s + (−0.135 − 0.505i)10-s + (−4.03 − 0.802i)11-s + (−1.56 + 3.09i)12-s + (−1.78 − 1.19i)13-s + (3.37 − 3.85i)14-s + (0.457 − 0.448i)15-s + (1.03 + 3.86i)16-s + (4.80 + 4.80i)17-s + ⋯
L(s)  = 1  + (0.946 + 0.321i)2-s + (0.186 + 0.982i)3-s + (0.793 + 0.609i)4-s + (−0.0918 − 0.137i)5-s + (−0.139 + 0.990i)6-s + (0.524 − 1.26i)7-s + (0.554 + 0.831i)8-s + (−0.930 + 0.365i)9-s + (−0.0427 − 0.159i)10-s + (−1.21 − 0.242i)11-s + (−0.450 + 0.892i)12-s + (−0.496 − 0.331i)13-s + (0.903 − 1.02i)14-s + (0.118 − 0.115i)15-s + (0.257 + 0.966i)16-s + (1.16 + 1.16i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.480 - 0.876i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.480 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.480 - 0.876i$
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: \((2,\ 192,\ (\ :1/2),\ 0.480 - 0.876i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73380 + 1.02663i\)
\(L(\frac12)\) \(\approx\) \(1.73380 + 1.02663i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.33 - 0.454i)T \)
3 \( 1 + (-0.322 - 1.70i)T \)
good5 \( 1 + (0.205 + 0.307i)T + (-1.91 + 4.61i)T^{2} \)
7 \( 1 + (-1.38 + 3.34i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (4.03 + 0.802i)T + (10.1 + 4.20i)T^{2} \)
13 \( 1 + (1.78 + 1.19i)T + (4.97 + 12.0i)T^{2} \)
17 \( 1 + (-4.80 - 4.80i)T + 17iT^{2} \)
19 \( 1 + (4.56 + 3.05i)T + (7.27 + 17.5i)T^{2} \)
23 \( 1 + (0.318 + 0.769i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (0.933 + 4.69i)T + (-26.7 + 11.0i)T^{2} \)
31 \( 1 - 3.86T + 31T^{2} \)
37 \( 1 + (-5.93 - 8.88i)T + (-14.1 + 34.1i)T^{2} \)
41 \( 1 + (2.81 - 1.16i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (-0.497 - 0.0989i)T + (39.7 + 16.4i)T^{2} \)
47 \( 1 + (3.50 - 3.50i)T - 47iT^{2} \)
53 \( 1 + (-0.878 + 4.41i)T + (-48.9 - 20.2i)T^{2} \)
59 \( 1 + (7.88 - 5.27i)T + (22.5 - 54.5i)T^{2} \)
61 \( 1 + (-1.65 - 8.31i)T + (-56.3 + 23.3i)T^{2} \)
67 \( 1 + (-0.680 + 0.135i)T + (61.8 - 25.6i)T^{2} \)
71 \( 1 + (7.27 + 3.01i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (6.01 - 2.49i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (-6.21 + 6.21i)T - 79iT^{2} \)
83 \( 1 + (-1.80 + 2.70i)T + (-31.7 - 76.6i)T^{2} \)
89 \( 1 + (-4.83 - 2.00i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 - 13.4iT - 97T^{2} \)
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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

−12.98983017690510165845760313840, −11.72619016956558020541202451404, −10.56689205803971688516784519572, −10.27811545192288565775805904994, −8.256629892764085295118987280052, −7.79082923741771865712683721081, −6.17310874901103197521899094295, −4.89612195145673213266241628551, −4.20026411068760715702691987256, −2.84028191028812425500649940225, 2.02612774597495383510207150508, 3.01094701712217556876134568220, 5.06036712173002681543864833817, 5.80569004459584738812704657165, 7.16169625488202192776851627020, 8.036283917455981369674327654444, 9.419392599113818552271125662674, 10.80302967452064396470122315537, 11.81405587629781479074239173023, 12.39556625177491395875188087989

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