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Properties

Label 2-117-117.103-c3-0-21
Degree $2$
Conductor $117$
Sign $0.570 - 0.821i$
Analytic cond. $6.90322$
Root an. cond. $2.62739$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.10 + 1.79i)2-s + (−2.81 + 4.36i)3-s + (2.42 − 4.20i)4-s + (6.91 + 3.99i)5-s + (0.917 − 18.6i)6-s + (30.1 − 17.4i)7-s − 11.2i·8-s + (−11.1 − 24.5i)9-s − 28.6·10-s + (5.70 − 3.29i)11-s + (11.5 + 22.4i)12-s + (43.8 − 16.6i)13-s + (−62.4 + 108. i)14-s + (−36.9 + 18.9i)15-s + (39.6 + 68.6i)16-s + 47.2·17-s + ⋯
L(s)  = 1  + (−1.09 + 0.633i)2-s + (−0.542 + 0.840i)3-s + (0.303 − 0.526i)4-s + (0.618 + 0.357i)5-s + (0.0624 − 1.26i)6-s + (1.62 − 0.940i)7-s − 0.497i·8-s + (−0.412 − 0.910i)9-s − 0.905·10-s + (0.156 − 0.0902i)11-s + (0.277 + 0.540i)12-s + (0.934 − 0.355i)13-s + (−1.19 + 2.06i)14-s + (−0.635 + 0.326i)15-s + (0.619 + 1.07i)16-s + 0.673·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.570 - 0.821i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.570 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.570 - 0.821i$
Analytic conductor: \(6.90322\)
Root analytic conductor: \(2.62739\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :3/2),\ 0.570 - 0.821i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.874257 + 0.456910i\)
\(L(\frac12)\) \(\approx\) \(0.874257 + 0.456910i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.81 - 4.36i)T \)
13 \( 1 + (-43.8 + 16.6i)T \)
good2 \( 1 + (3.10 - 1.79i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + (-6.91 - 3.99i)T + (62.5 + 108. i)T^{2} \)
7 \( 1 + (-30.1 + 17.4i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (-5.70 + 3.29i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 - 47.2T + 4.91e3T^{2} \)
19 \( 1 + 103. iT - 6.85e3T^{2} \)
23 \( 1 + (56.0 - 97.0i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-33.7 - 58.3i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-54.7 - 31.6i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 61.7iT - 5.06e4T^{2} \)
41 \( 1 + (81.9 + 47.3i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (83.1 + 143. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-50.8 + 29.3i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 558.T + 1.48e5T^{2} \)
59 \( 1 + (-647. - 373. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-449. - 778. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-121. - 70.1i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 936. iT - 3.57e5T^{2} \)
73 \( 1 - 650. iT - 3.89e5T^{2} \)
79 \( 1 + (-435. - 754. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-549. + 317. i)T + (2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 509. iT - 7.04e5T^{2} \)
97 \( 1 + (1.04e3 - 601. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−13.51035190068014586558732955884, −11.67285321620522594443958432682, −10.72556151102918388993820339964, −10.10416729074628206068207411602, −8.906319918244961834395253145304, −7.918157352146810837980134746629, −6.70818917703581297356040723439, −5.37917553845130708112993535074, −3.92992493730267229506658694634, −1.03294972896963764420180924700, 1.32194660200368243276410983149, 2.03354304204631086621037971750, 5.10453947922151532630720750828, 6.04730981547702647510730369602, 8.047837931652757198882889323237, 8.396389709409170602084524918888, 9.734090136785288232614176683085, 10.96080671394544510153360263531, 11.65892544326186630212525287470, 12.43533894255521278354639309469

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