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Properties

Label 2-117-9.5-c2-0-16
Degree $2$
Conductor $117$
Sign $-0.629 + 0.776i$
Analytic cond. $3.18801$
Root an. cond. $1.78550$
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  + (−3.22 + 1.86i)2-s + (0.496 − 2.95i)3-s + (4.93 − 8.54i)4-s + (0.733 + 0.423i)5-s + (3.90 + 10.4i)6-s + (−2.32 − 4.01i)7-s + 21.8i·8-s + (−8.50 − 2.93i)9-s − 3.15·10-s + (−10.6 + 6.16i)11-s + (−22.8 − 18.8i)12-s + (−1.80 + 3.12i)13-s + (14.9 + 8.63i)14-s + (1.61 − 1.96i)15-s + (−20.9 − 36.2i)16-s − 25.9i·17-s + ⋯
L(s)  = 1  + (−1.61 + 0.930i)2-s + (0.165 − 0.986i)3-s + (1.23 − 2.13i)4-s + (0.146 + 0.0847i)5-s + (0.651 + 1.74i)6-s + (−0.331 − 0.574i)7-s + 2.73i·8-s + (−0.945 − 0.326i)9-s − 0.315·10-s + (−0.970 + 0.560i)11-s + (−1.90 − 1.56i)12-s + (−0.138 + 0.240i)13-s + (1.06 + 0.617i)14-s + (0.107 − 0.130i)15-s + (−1.30 − 2.26i)16-s − 1.52i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.629 + 0.776i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.629 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.629 + 0.776i$
Analytic conductor: \(3.18801\)
Root analytic conductor: \(1.78550\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1),\ -0.629 + 0.776i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.123980 - 0.260159i\)
\(L(\frac12)\) \(\approx\) \(0.123980 - 0.260159i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.496 + 2.95i)T \)
13 \( 1 + (1.80 - 3.12i)T \)
good2 \( 1 + (3.22 - 1.86i)T + (2 - 3.46i)T^{2} \)
5 \( 1 + (-0.733 - 0.423i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (2.32 + 4.01i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (10.6 - 6.16i)T + (60.5 - 104. i)T^{2} \)
17 \( 1 + 25.9iT - 289T^{2} \)
19 \( 1 + 12.4T + 361T^{2} \)
23 \( 1 + (-0.958 - 0.553i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (40.9 - 23.6i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-5.53 + 9.57i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 4.57T + 1.36e3T^{2} \)
41 \( 1 + (64.3 + 37.1i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-35.0 - 60.6i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-61.9 + 35.7i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 88.9iT - 2.80e3T^{2} \)
59 \( 1 + (-35.0 - 20.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-29.6 - 51.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-11.8 + 20.5i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 49.7iT - 5.04e3T^{2} \)
73 \( 1 - 10.8T + 5.32e3T^{2} \)
79 \( 1 + (3.54 + 6.13i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-14.0 + 8.12i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 31.1iT - 7.92e3T^{2} \)
97 \( 1 + (50.9 + 88.3i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−13.09272702501050001521765423542, −11.61537163505527065814025259197, −10.43171466232481415471789801602, −9.501482206345124253927602924283, −8.406308601124964929815843634085, −7.34662348328921715678831028908, −6.87966183936390884572052520286, −5.51099735199299567248936870723, −2.21252598026336339194040896378, −0.31051741515279282621295833119, 2.35031573048095493662594938460, 3.65349316292417534809230668172, 5.81354095803711066459364550684, 7.83535571128855652578464164568, 8.664873066810216427468746708056, 9.496834204690298509979137801990, 10.47751103236856546901472420569, 11.00861079911640623116216775424, 12.24904345565588201305269633195, 13.29204858667899789211492765726

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