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Properties

Label 2-117-117.103-c3-0-13
Degree $2$
Conductor $117$
Sign $-0.437 - 0.898i$
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  + (−4.76 + 2.75i)2-s + (2.56 + 4.51i)3-s + (11.1 − 19.3i)4-s + (−10.5 − 6.07i)5-s + (−24.6 − 14.4i)6-s + (15.9 − 9.18i)7-s + 78.7i·8-s + (−13.7 + 23.2i)9-s + 66.9·10-s + (21.1 − 12.2i)11-s + (115. + 0.746i)12-s + (34.1 + 32.1i)13-s + (−50.5 + 87.6i)14-s + (0.406 − 63.1i)15-s + (−127. − 221. i)16-s + 36.2·17-s + ⋯
L(s)  = 1  + (−1.68 + 0.973i)2-s + (0.494 + 0.869i)3-s + (1.39 − 2.41i)4-s + (−0.941 − 0.543i)5-s + (−1.67 − 0.984i)6-s + (0.859 − 0.496i)7-s + 3.48i·8-s + (−0.511 + 0.859i)9-s + 2.11·10-s + (0.579 − 0.334i)11-s + (2.78 + 0.0179i)12-s + (0.727 + 0.685i)13-s + (−0.965 + 1.67i)14-s + (0.00699 − 1.08i)15-s + (−1.99 − 3.45i)16-s + 0.516·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.437 - 0.898i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.437 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.437 - 0.898i$
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: \((2,\ 117,\ (\ :3/2),\ -0.437 - 0.898i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.408519 + 0.653462i\)
\(L(\frac12)\) \(\approx\) \(0.408519 + 0.653462i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.56 - 4.51i)T \)
13 \( 1 + (-34.1 - 32.1i)T \)
good2 \( 1 + (4.76 - 2.75i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + (10.5 + 6.07i)T + (62.5 + 108. i)T^{2} \)
7 \( 1 + (-15.9 + 9.18i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (-21.1 + 12.2i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 - 36.2T + 4.91e3T^{2} \)
19 \( 1 - 139. iT - 6.85e3T^{2} \)
23 \( 1 + (-4.44 + 7.69i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-107. - 185. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (67.1 + 38.7i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 94.1iT - 5.06e4T^{2} \)
41 \( 1 + (-26.0 - 15.0i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-132. - 229. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-54.3 + 31.3i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 555.T + 1.48e5T^{2} \)
59 \( 1 + (-126. - 73.0i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-386. - 669. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (19.1 + 11.0i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 820. iT - 3.57e5T^{2} \)
73 \( 1 - 267. iT - 3.89e5T^{2} \)
79 \( 1 + (-66.9 - 115. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (239. - 138. i)T + (2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 1.18e3iT - 7.04e5T^{2} \)
97 \( 1 + (-717. + 414. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−14.26177831896489855545998399226, −11.72320084346979195380267530687, −10.89400043425389242511613774619, −9.951093751074647525975172046450, −8.722102373882474312902696310551, −8.308894048281793279206657796146, −7.34736167537964420953470544222, −5.68922164725327720100701091880, −4.15685211094494839758935353125, −1.30923756946331458249939732716, 0.797268810851397958964386931994, 2.33184983485982912835052594129, 3.56357312798628862740021812271, 6.78304283198260873249937119898, 7.71931207697488819906642542614, 8.397417716040688180873410296315, 9.298353314710506869195835004309, 10.78072801364124663375609600769, 11.60933067133331007205093670169, 12.10543260121194620948394035004

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