Properties

Label 2-117-117.103-c3-0-30
Degree $2$
Conductor $117$
Sign $-0.219 + 0.975i$
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  + (−1.12 + 0.649i)2-s + (−1.28 − 5.03i)3-s + (−3.15 + 5.46i)4-s + (4.00 + 2.31i)5-s + (4.71 + 4.83i)6-s + (7.19 − 4.15i)7-s − 18.5i·8-s + (−23.7 + 12.9i)9-s − 6.01·10-s + (32.0 − 18.5i)11-s + (31.5 + 8.87i)12-s + (−29.3 − 36.5i)13-s + (−5.39 + 9.35i)14-s + (6.51 − 23.1i)15-s + (−13.1 − 22.7i)16-s − 98.9·17-s + ⋯
L(s)  = 1  + (−0.397 + 0.229i)2-s + (−0.246 − 0.969i)3-s + (−0.394 + 0.683i)4-s + (0.358 + 0.207i)5-s + (0.320 + 0.328i)6-s + (0.388 − 0.224i)7-s − 0.821i·8-s + (−0.878 + 0.478i)9-s − 0.190·10-s + (0.879 − 0.507i)11-s + (0.759 + 0.213i)12-s + (−0.626 − 0.779i)13-s + (−0.103 + 0.178i)14-s + (0.112 − 0.398i)15-s + (−0.205 − 0.356i)16-s − 1.41·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.219 + 0.975i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.219 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.219 + 0.975i$
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: \((2,\ 117,\ (\ :3/2),\ -0.219 + 0.975i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.512673 - 0.640506i\)
\(L(\frac12)\) \(\approx\) \(0.512673 - 0.640506i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.28 + 5.03i)T \)
13 \( 1 + (29.3 + 36.5i)T \)
good2 \( 1 + (1.12 - 0.649i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + (-4.00 - 2.31i)T + (62.5 + 108. i)T^{2} \)
7 \( 1 + (-7.19 + 4.15i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (-32.0 + 18.5i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + 98.9T + 4.91e3T^{2} \)
19 \( 1 + 138. iT - 6.85e3T^{2} \)
23 \( 1 + (-35.5 + 61.5i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-1.75 - 3.03i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-109. - 63.2i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 256. iT - 5.06e4T^{2} \)
41 \( 1 + (-390. - 225. i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (202. + 351. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (498. - 287. i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 275.T + 1.48e5T^{2} \)
59 \( 1 + (-177. - 102. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-171. - 297. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (451. + 260. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 294. iT - 3.57e5T^{2} \)
73 \( 1 + 539. iT - 3.89e5T^{2} \)
79 \( 1 + (-234. - 406. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-299. + 173. i)T + (2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 1.09e3iT - 7.04e5T^{2} \)
97 \( 1 + (-482. + 278. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−12.91008667824260735005345816890, −11.79402365917696337478426651756, −10.81974910737725474607213866915, −9.216193386701832400374863304987, −8.359022827574142830265599132901, −7.22960243351084345912853212357, −6.38541797672357761993813445659, −4.63352842330652065191379632316, −2.66134860320478708184315275504, −0.52172006425934288548197066794, 1.79522841028214244102416612174, 4.19512824532124730431450442121, 5.18819244409227129208346535572, 6.40487727956605216829735748214, 8.432723990413200880639305291709, 9.459200181558258722928864588363, 9.887732177665733089398764490632, 11.18048908936502595087444265771, 11.89140029642632335110963051978, 13.54722783258645350193968886859

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