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Properties

Label 2-117-13.3-c3-0-0
Degree $2$
Conductor $117$
Sign $-0.218 - 0.975i$
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  + (−1.37 − 2.38i)2-s + (0.201 − 0.348i)4-s + 0.313·5-s + (−14.2 + 24.7i)7-s − 23.1·8-s + (−0.431 − 0.747i)10-s + (−31.6 − 54.7i)11-s + (−2.26 + 46.8i)13-s + 78.7·14-s + (30.3 + 52.4i)16-s + (−49.4 + 85.5i)17-s + (7.24 − 12.5i)19-s + (0.0631 − 0.109i)20-s + (−87.1 + 150. i)22-s + (7.19 + 12.4i)23-s + ⋯
L(s)  = 1  + (−0.487 − 0.843i)2-s + (0.0251 − 0.0436i)4-s + 0.0280·5-s + (−0.771 + 1.33i)7-s − 1.02·8-s + (−0.0136 − 0.0236i)10-s + (−0.866 − 1.50i)11-s + (−0.0484 + 0.998i)13-s + 1.50·14-s + (0.473 + 0.820i)16-s + (−0.704 + 1.22i)17-s + (0.0874 − 0.151i)19-s + (0.000705 − 0.00122i)20-s + (−0.844 + 1.46i)22-s + (0.0652 + 0.112i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.218 - 0.975i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.218 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.218 - 0.975i$
Analytic conductor: \(6.90322\)
Root analytic conductor: \(2.62739\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :3/2),\ -0.218 - 0.975i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.115503 + 0.144150i\)
\(L(\frac12)\) \(\approx\) \(0.115503 + 0.144150i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (2.26 - 46.8i)T \)
good2 \( 1 + (1.37 + 2.38i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 - 0.313T + 125T^{2} \)
7 \( 1 + (14.2 - 24.7i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (31.6 + 54.7i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (49.4 - 85.5i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-7.24 + 12.5i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-7.19 - 12.4i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-98.1 - 169. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 118.T + 2.97e4T^{2} \)
37 \( 1 + (159. + 276. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (173. + 299. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-34.7 + 60.1i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 101.T + 1.03e5T^{2} \)
53 \( 1 + 594.T + 1.48e5T^{2} \)
59 \( 1 + (-102. + 177. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-107. + 187. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (34.3 + 59.4i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (473. - 819. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 779.T + 3.89e5T^{2} \)
79 \( 1 - 240.T + 4.93e5T^{2} \)
83 \( 1 - 855.T + 5.71e5T^{2} \)
89 \( 1 + (632. + 1.09e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (331. - 573. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−13.04363736682678622176867270245, −12.10842952063721677620472169043, −11.20927990179698036223614637170, −10.30287483563413923790641111592, −9.084100844743465005991224757521, −8.536065001354021608427471008940, −6.42407265827843730169787644217, −5.60538785567949880939035108398, −3.31199662251078439709500016483, −2.08864406238712216236794757871, 0.10571199121443442236040288347, 2.94848235936147663030206406582, 4.69926339873580541248930955870, 6.42011520357943486421431801806, 7.33049222102304164515449763096, 8.016685270799123270701140999227, 9.681573472184637393995756972078, 10.21578425931974780868322110214, 11.79736874096178713599021170309, 12.93462816367178668260922854930

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