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Properties

Label 2-141-47.46-c6-0-9
Degree $2$
Conductor $141$
Sign $-0.733 - 0.680i$
Analytic cond. $32.4376$
Root an. cond. $5.69540$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 13.9·2-s + 15.5·3-s + 131.·4-s + 78.4i·5-s − 218.·6-s + 344.·7-s − 947.·8-s + 243·9-s − 1.09e3i·10-s + 1.78e3i·11-s + 2.05e3·12-s + 2.16e3i·13-s − 4.81e3·14-s + 1.22e3i·15-s + 4.82e3·16-s − 6.01e3·17-s + ⋯
L(s)  = 1  − 1.74·2-s + 0.577·3-s + 2.05·4-s + 0.627i·5-s − 1.00·6-s + 1.00·7-s − 1.85·8-s + 0.333·9-s − 1.09i·10-s + 1.33i·11-s + 1.18·12-s + 0.984i·13-s − 1.75·14-s + 0.362i·15-s + 1.17·16-s − 1.22·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 141 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.733 - 0.680i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.733 - 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(141\)    =    \(3 \cdot 47\)
Sign: $-0.733 - 0.680i$
Analytic conductor: \(32.4376\)
Root analytic conductor: \(5.69540\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{141} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 141,\ (\ :3),\ -0.733 - 0.680i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.8285549695\)
\(L(\frac12)\) \(\approx\) \(0.8285549695\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 15.5T \)
47 \( 1 + (7.61e4 + 7.06e4i)T \)
good2 \( 1 + 13.9T + 64T^{2} \)
5 \( 1 - 78.4iT - 1.56e4T^{2} \)
7 \( 1 - 344.T + 1.17e5T^{2} \)
11 \( 1 - 1.78e3iT - 1.77e6T^{2} \)
13 \( 1 - 2.16e3iT - 4.82e6T^{2} \)
17 \( 1 + 6.01e3T + 2.41e7T^{2} \)
19 \( 1 - 5.04e3iT - 4.70e7T^{2} \)
23 \( 1 - 1.05e4iT - 1.48e8T^{2} \)
29 \( 1 + 2.88e4iT - 5.94e8T^{2} \)
31 \( 1 + 4.14e4iT - 8.87e8T^{2} \)
37 \( 1 + 8.84e3T + 2.56e9T^{2} \)
41 \( 1 - 1.36e5iT - 4.75e9T^{2} \)
43 \( 1 + 7.70e4iT - 6.32e9T^{2} \)
53 \( 1 + 1.21e4T + 2.21e10T^{2} \)
59 \( 1 - 2.63e4T + 4.21e10T^{2} \)
61 \( 1 + 1.09e5T + 5.15e10T^{2} \)
67 \( 1 - 3.72e5iT - 9.04e10T^{2} \)
71 \( 1 + 5.99e5T + 1.28e11T^{2} \)
73 \( 1 - 1.27e5iT - 1.51e11T^{2} \)
79 \( 1 - 7.40e5T + 2.43e11T^{2} \)
83 \( 1 - 5.75e5T + 3.26e11T^{2} \)
89 \( 1 - 9.35e4T + 4.96e11T^{2} \)
97 \( 1 + 1.03e6T + 8.32e11T^{2} \)
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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

−11.81277382317928373303589315649, −11.14468887555591233510350203129, −10.02555057920789941875220090947, −9.334147318013352232381342432089, −8.240501658366256740933506721315, −7.44725055252919415506340319022, −6.58422855124842527049613234494, −4.38238357951205505346579026645, −2.31956206629555790164245117874, −1.61155576010649194443794827645, 0.42175071415912678100624191376, 1.47229388226647093793885271094, 2.89046867461337430194248246816, 4.96223300410196310707671285184, 6.67711069798032071261335989753, 7.913117445889395630317291430591, 8.677948655106781708172027340041, 9.010515376764784760837076989145, 10.68788744669951439432047602640, 10.96460767320743661258500985187

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