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Properties

Label 2-192-192.131-c1-0-25
Degree $2$
Conductor $192$
Sign $0.606 + 0.795i$
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  + (1.40 − 0.166i)2-s + (−1.62 − 0.593i)3-s + (1.94 − 0.466i)4-s + (0.720 − 1.07i)5-s + (−2.38 − 0.563i)6-s + (−0.767 − 1.85i)7-s + (2.65 − 0.978i)8-s + (2.29 + 1.93i)9-s + (0.833 − 1.63i)10-s + (−0.967 + 0.192i)11-s + (−3.44 − 0.395i)12-s + (2.76 − 1.84i)13-s + (−1.38 − 2.47i)14-s + (−1.81 + 1.32i)15-s + (3.56 − 1.81i)16-s + (−4.20 + 4.20i)17-s + ⋯
L(s)  = 1  + (0.993 − 0.117i)2-s + (−0.939 − 0.342i)3-s + (0.972 − 0.233i)4-s + (0.322 − 0.482i)5-s + (−0.973 − 0.229i)6-s + (−0.289 − 0.699i)7-s + (0.938 − 0.345i)8-s + (0.765 + 0.643i)9-s + (0.263 − 0.516i)10-s + (−0.291 + 0.0580i)11-s + (−0.993 − 0.114i)12-s + (0.767 − 0.512i)13-s + (−0.370 − 0.661i)14-s + (−0.468 + 0.342i)15-s + (0.891 − 0.453i)16-s + (−1.01 + 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.606 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.606 + 0.795i$
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: \((2,\ 192,\ (\ :1/2),\ 0.606 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48969 - 0.737815i\)
\(L(\frac12)\) \(\approx\) \(1.48969 - 0.737815i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 + 0.166i)T \)
3 \( 1 + (1.62 + 0.593i)T \)
good5 \( 1 + (-0.720 + 1.07i)T + (-1.91 - 4.61i)T^{2} \)
7 \( 1 + (0.767 + 1.85i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (0.967 - 0.192i)T + (10.1 - 4.20i)T^{2} \)
13 \( 1 + (-2.76 + 1.84i)T + (4.97 - 12.0i)T^{2} \)
17 \( 1 + (4.20 - 4.20i)T - 17iT^{2} \)
19 \( 1 + (-1.31 + 0.879i)T + (7.27 - 17.5i)T^{2} \)
23 \( 1 + (3.41 - 8.25i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-0.303 + 1.52i)T + (-26.7 - 11.0i)T^{2} \)
31 \( 1 - 1.85T + 31T^{2} \)
37 \( 1 + (3.53 - 5.29i)T + (-14.1 - 34.1i)T^{2} \)
41 \( 1 + (0.606 + 0.251i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (7.79 - 1.54i)T + (39.7 - 16.4i)T^{2} \)
47 \( 1 + (-3.56 - 3.56i)T + 47iT^{2} \)
53 \( 1 + (-0.605 - 3.04i)T + (-48.9 + 20.2i)T^{2} \)
59 \( 1 + (6.01 + 4.01i)T + (22.5 + 54.5i)T^{2} \)
61 \( 1 + (-2.31 + 11.6i)T + (-56.3 - 23.3i)T^{2} \)
67 \( 1 + (-12.2 - 2.43i)T + (61.8 + 25.6i)T^{2} \)
71 \( 1 + (-10.2 + 4.23i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (8.22 + 3.40i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (-4.22 - 4.22i)T + 79iT^{2} \)
83 \( 1 + (8.63 + 12.9i)T + (-31.7 + 76.6i)T^{2} \)
89 \( 1 + (10.4 - 4.32i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 + 4.02iT - 97T^{2} \)
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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

−12.59971552982382744731560708388, −11.49758145867779592028254966527, −10.76347830799372217109012397325, −9.851092067198715640923864564087, −8.003998255621574504767296000788, −6.84410689260906250679827416248, −5.93099687173723650305129885509, −4.97931698797219112214821618700, −3.70880129500629187579853141200, −1.54811877853175509554612031903, 2.50863543334676837693087700900, 4.12377871791821160518405059750, 5.27840139971959648657240180234, 6.30162787720886154846412302201, 6.90307549859749017459702357656, 8.690372640906882708119334375575, 10.12542939820177143128114088157, 10.93956893941462348078806613841, 11.81692218556460653933829662755, 12.55268449759154834395220834793

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