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Properties

Label 2-192-192.131-c1-0-7
Degree $2$
Conductor $192$
Sign $0.902 - 0.430i$
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  + (−0.484 − 1.32i)2-s + (1.60 + 0.640i)3-s + (−1.52 + 1.28i)4-s + (−2.22 + 3.33i)5-s + (0.0708 − 2.44i)6-s + (0.140 + 0.338i)7-s + (2.45 + 1.40i)8-s + (2.17 + 2.06i)9-s + (5.51 + 1.34i)10-s + (4.16 − 0.827i)11-s + (−3.28 + 1.09i)12-s + (−1.81 + 1.21i)13-s + (0.381 − 0.350i)14-s + (−5.72 + 3.93i)15-s + (0.679 − 3.94i)16-s + (2.34 − 2.34i)17-s + ⋯
L(s)  = 1  + (−0.342 − 0.939i)2-s + (0.929 + 0.369i)3-s + (−0.764 + 0.644i)4-s + (−0.996 + 1.49i)5-s + (0.0289 − 0.999i)6-s + (0.0530 + 0.127i)7-s + (0.867 + 0.497i)8-s + (0.726 + 0.687i)9-s + (1.74 + 0.424i)10-s + (1.25 − 0.249i)11-s + (−0.948 + 0.315i)12-s + (−0.502 + 0.335i)13-s + (0.102 − 0.0936i)14-s + (−1.47 + 1.01i)15-s + (0.169 − 0.985i)16-s + (0.568 − 0.568i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 - 0.430i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.902 - 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.902 - 0.430i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1/2),\ 0.902 - 0.430i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07113 + 0.242327i\)
\(L(\frac12)\) \(\approx\) \(1.07113 + 0.242327i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.484 + 1.32i)T \)
3 \( 1 + (-1.60 - 0.640i)T \)
good5 \( 1 + (2.22 - 3.33i)T + (-1.91 - 4.61i)T^{2} \)
7 \( 1 + (-0.140 - 0.338i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (-4.16 + 0.827i)T + (10.1 - 4.20i)T^{2} \)
13 \( 1 + (1.81 - 1.21i)T + (4.97 - 12.0i)T^{2} \)
17 \( 1 + (-2.34 + 2.34i)T - 17iT^{2} \)
19 \( 1 + (5.58 - 3.73i)T + (7.27 - 17.5i)T^{2} \)
23 \( 1 + (-0.561 + 1.35i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-0.511 + 2.57i)T + (-26.7 - 11.0i)T^{2} \)
31 \( 1 - 7.17T + 31T^{2} \)
37 \( 1 + (-2.65 + 3.96i)T + (-14.1 - 34.1i)T^{2} \)
41 \( 1 + (5.19 + 2.15i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (0.439 - 0.0874i)T + (39.7 - 16.4i)T^{2} \)
47 \( 1 + (2.23 + 2.23i)T + 47iT^{2} \)
53 \( 1 + (-1.03 - 5.20i)T + (-48.9 + 20.2i)T^{2} \)
59 \( 1 + (-6.44 - 4.30i)T + (22.5 + 54.5i)T^{2} \)
61 \( 1 + (-2.26 + 11.3i)T + (-56.3 - 23.3i)T^{2} \)
67 \( 1 + (-3.80 - 0.757i)T + (61.8 + 25.6i)T^{2} \)
71 \( 1 + (3.54 - 1.46i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (8.70 + 3.60i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (-2.50 - 2.50i)T + 79iT^{2} \)
83 \( 1 + (-1.65 - 2.47i)T + (-31.7 + 76.6i)T^{2} \)
89 \( 1 + (-11.1 + 4.63i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 - 10.0iT - 97T^{2} \)
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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

−12.23345121355053424858621065068, −11.57372043233498268143700000988, −10.55926644737200548165216428120, −9.845664812116444929788002939253, −8.671459820998745397288459823905, −7.80857013049372056369671377294, −6.76246499135391796627019745009, −4.28268735740172769187473777388, −3.52975963502559652439799479946, −2.38987182269131993094889019322, 1.15471027267562869343569244343, 3.95854104276884693648822441864, 4.78862226155813301361072506640, 6.53157646123002253263776708292, 7.60669000946888338830043092271, 8.459444200369320991318329795117, 8.957781702438854146937350216236, 10.00000501523976588446879208256, 11.80781302300874615769395233377, 12.71580284416184857791034814475

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