Properties

Label 2-192-192.59-c1-0-25
Degree $2$
Conductor $192$
Sign $-0.969 + 0.245i$
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  + (0.246 − 1.39i)2-s + (−1.19 + 1.25i)3-s + (−1.87 − 0.687i)4-s + (0.359 − 1.80i)5-s + (1.45 + 1.97i)6-s + (−3.05 − 1.26i)7-s + (−1.42 + 2.44i)8-s + (−0.144 − 2.99i)9-s + (−2.42 − 0.946i)10-s + (−2.05 − 3.07i)11-s + (3.10 − 1.53i)12-s + (0.534 − 0.106i)13-s + (−2.51 + 3.93i)14-s + (1.83 + 2.60i)15-s + (3.05 + 2.58i)16-s + (−0.827 − 0.827i)17-s + ⋯
L(s)  = 1  + (0.174 − 0.984i)2-s + (−0.689 + 0.723i)3-s + (−0.939 − 0.343i)4-s + (0.160 − 0.807i)5-s + (0.592 + 0.805i)6-s + (−1.15 − 0.477i)7-s + (−0.502 + 0.864i)8-s + (−0.0481 − 0.998i)9-s + (−0.767 − 0.299i)10-s + (−0.620 − 0.928i)11-s + (0.896 − 0.442i)12-s + (0.148 − 0.0295i)13-s + (−0.671 + 1.05i)14-s + (0.474 + 0.673i)15-s + (0.763 + 0.645i)16-s + (−0.200 − 0.200i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.245i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 + 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-0.969 + 0.245i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1/2),\ -0.969 + 0.245i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0708130 - 0.567744i\)
\(L(\frac12)\) \(\approx\) \(0.0708130 - 0.567744i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.246 + 1.39i)T \)
3 \( 1 + (1.19 - 1.25i)T \)
good5 \( 1 + (-0.359 + 1.80i)T + (-4.61 - 1.91i)T^{2} \)
7 \( 1 + (3.05 + 1.26i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (2.05 + 3.07i)T + (-4.20 + 10.1i)T^{2} \)
13 \( 1 + (-0.534 + 0.106i)T + (12.0 - 4.97i)T^{2} \)
17 \( 1 + (0.827 + 0.827i)T + 17iT^{2} \)
19 \( 1 + (6.44 - 1.28i)T + (17.5 - 7.27i)T^{2} \)
23 \( 1 + (-7.47 + 3.09i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (0.420 + 0.280i)T + (11.0 + 26.7i)T^{2} \)
31 \( 1 - 2.79T + 31T^{2} \)
37 \( 1 + (0.213 - 1.07i)T + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (3.82 + 9.23i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (-3.05 - 4.56i)T + (-16.4 + 39.7i)T^{2} \)
47 \( 1 + (-7.47 + 7.47i)T - 47iT^{2} \)
53 \( 1 + (2.02 - 1.35i)T + (20.2 - 48.9i)T^{2} \)
59 \( 1 + (10.5 + 2.10i)T + (54.5 + 22.5i)T^{2} \)
61 \( 1 + (-0.153 - 0.102i)T + (23.3 + 56.3i)T^{2} \)
67 \( 1 + (-1.39 + 2.08i)T + (-25.6 - 61.8i)T^{2} \)
71 \( 1 + (5.26 - 12.7i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (3.57 + 8.64i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-9.99 + 9.99i)T - 79iT^{2} \)
83 \( 1 + (1.76 + 8.89i)T + (-76.6 + 31.7i)T^{2} \)
89 \( 1 + (4.17 - 10.0i)T + (-62.9 - 62.9i)T^{2} \)
97 \( 1 - 15.1iT - 97T^{2} \)
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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

−12.15444501075018853718957555141, −10.80269916133724703428325176955, −10.49215023651010280322369288445, −9.273230092696442127363715366840, −8.641734382313066547132296579202, −6.47481336168153406974257881339, −5.37733404662586984023128685899, −4.31920557902639108997079680749, −3.12261528573455493900805664531, −0.51333749197481787668830085147, 2.80378124780662789824424677439, 4.73712513841728877312813882894, 6.04136107315654423631092671127, 6.68277150019395557600196229149, 7.45508694668020007664068419549, 8.814541393100970903920681301879, 10.00736895617616560014035749312, 11.00944882476726454656972505863, 12.58218198772495124011949307358, 12.83425806279559722015445051595

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