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Properties

Label 2-171-19.12-c2-0-7
Degree $2$
Conductor $171$
Sign $0.942 - 0.334i$
Analytic cond. $4.65941$
Root an. cond. $2.15856$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.78 − 1.03i)2-s + (0.131 − 0.227i)4-s + (3.20 + 5.55i)5-s − 2.26·7-s + 7.71i·8-s + (11.4 + 6.62i)10-s + 20.0·11-s + (−0.135 − 0.0782i)13-s + (−4.04 + 2.33i)14-s + (8.49 + 14.7i)16-s + (−12.3 − 21.3i)17-s + (−18.9 − 1.60i)19-s + 1.68·20-s + (35.9 − 20.7i)22-s + (2.62 − 4.54i)23-s + ⋯
L(s)  = 1  + (0.894 − 0.516i)2-s + (0.0328 − 0.0569i)4-s + (0.641 + 1.11i)5-s − 0.323·7-s + 0.964i·8-s + (1.14 + 0.662i)10-s + 1.82·11-s + (−0.0104 − 0.00601i)13-s + (−0.289 + 0.166i)14-s + (0.530 + 0.919i)16-s + (−0.724 − 1.25i)17-s + (−0.996 − 0.0847i)19-s + 0.0842·20-s + (1.63 − 0.942i)22-s + (0.114 − 0.197i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.334i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.942 - 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.942 - 0.334i$
Analytic conductor: \(4.65941\)
Root analytic conductor: \(2.15856\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1),\ 0.942 - 0.334i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.38582 + 0.410644i\)
\(L(\frac12)\) \(\approx\) \(2.38582 + 0.410644i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 + (18.9 + 1.60i)T \)
good2 \( 1 + (-1.78 + 1.03i)T + (2 - 3.46i)T^{2} \)
5 \( 1 + (-3.20 - 5.55i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + 2.26T + 49T^{2} \)
11 \( 1 - 20.0T + 121T^{2} \)
13 \( 1 + (0.135 + 0.0782i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (12.3 + 21.3i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (-2.62 + 4.54i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-31.4 - 18.1i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + 17.1iT - 961T^{2} \)
37 \( 1 + 42.7iT - 1.36e3T^{2} \)
41 \( 1 + (30.0 - 17.3i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-12.5 - 21.7i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-14.6 + 25.4i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (48.4 + 27.9i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-29.9 + 17.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-27.3 + 47.4i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-66.0 - 38.1i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (63.6 - 36.7i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (45.9 + 79.5i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (53.1 - 30.6i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 148.T + 6.88e3T^{2} \)
89 \( 1 + (-62.7 - 36.2i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (70.0 - 40.4i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−12.61936163002274418324443268837, −11.61379004806787630094680278287, −10.93858361275842275317416565145, −9.692505839261601012464438947420, −8.686528772061653760113866700483, −6.91798031712289855817003945282, −6.23112506437317849647053572906, −4.63451036551936690133105561734, −3.43463996699631144846193794621, −2.26517308622416227869612254818, 1.38719212027469824002866098683, 3.90604710589915128572488345485, 4.76985244951054493374735041850, 6.11725889634982985596265809278, 6.61245850292506738233149908610, 8.565386195116819819421697858545, 9.297507620969592115062987961805, 10.33374729214920751608966211038, 11.93616559651852556966704864089, 12.75290969391327833022199170268

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