↑

Properties

Label 2-117-117.38-c2-0-16
Degree $2$
Conductor $117$
Sign $0.503 + 0.864i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.04 + 1.80i)2-s + (−2.79 + 1.10i)3-s + (−0.168 − 0.292i)4-s + (−2.59 − 4.48i)5-s + (0.919 − 6.17i)6-s + (−1.68 − 0.970i)7-s − 7.62·8-s + (6.57 − 6.14i)9-s + 10.7·10-s + (5.82 − 10.0i)11-s + (0.792 + 0.629i)12-s + (−0.650 + 12.9i)13-s + (3.50 − 2.02i)14-s + (12.1 + 9.66i)15-s + (8.61 − 14.9i)16-s − 23.0i·17-s + ⋯
L(s)  = 1  + (−0.520 + 0.901i)2-s + (−0.930 + 0.367i)3-s + (−0.0421 − 0.0730i)4-s + (−0.518 − 0.897i)5-s + (0.153 − 1.02i)6-s + (−0.240 − 0.138i)7-s − 0.953·8-s + (0.730 − 0.682i)9-s + 1.07·10-s + (0.529 − 0.916i)11-s + (0.0660 + 0.0524i)12-s + (−0.0500 + 0.998i)13-s + (0.250 − 0.144i)14-s + (0.811 + 0.644i)15-s + (0.538 − 0.932i)16-s − 1.35i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.503 + 0.864i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.503 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.503 + 0.864i$
Analytic conductor: \(3.18801\)
Root analytic conductor: \(1.78550\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1),\ 0.503 + 0.864i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.296614 - 0.170539i\)
\(L(\frac12)\) \(\approx\) \(0.296614 - 0.170539i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.79 - 1.10i)T \)
13 \( 1 + (0.650 - 12.9i)T \)
good2 \( 1 + (1.04 - 1.80i)T + (-2 - 3.46i)T^{2} \)
5 \( 1 + (2.59 + 4.48i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (1.68 + 0.970i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-5.82 + 10.0i)T + (-60.5 - 104. i)T^{2} \)
17 \( 1 + 23.0iT - 289T^{2} \)
19 \( 1 + 15.9iT - 361T^{2} \)
23 \( 1 + (19.8 - 11.4i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (38.5 + 22.2i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (29.2 - 16.8i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + 36.7iT - 1.36e3T^{2} \)
41 \( 1 + (-1.66 - 2.87i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-4.62 + 8.01i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (35.1 - 60.8i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 36.5iT - 2.80e3T^{2} \)
59 \( 1 + (-42.5 - 73.6i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-16.2 + 28.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-21.7 + 12.5i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 16.8T + 5.04e3T^{2} \)
73 \( 1 - 82.1iT - 5.32e3T^{2} \)
79 \( 1 + (-6.27 + 10.8i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-62.7 + 108. i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 13.3T + 7.92e3T^{2} \)
97 \( 1 + (142. + 82.0i)T + (4.70e3 + 8.14e3i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.97277577995339464179259165489, −11.78423569209036370875961436800, −11.38052781830595342027815318894, −9.497527308987389917657465864821, −8.922198873386393280672083464693, −7.49979644160571246060582074648, −6.50790346050816927335457400640, −5.31323894487987109035948510081, −3.88497976314440918367664954970, −0.31670921944380222840600499132, 1.81963554887913810167486834707, 3.66564952024998465664726464793, 5.73544668233728450669171585868, 6.75779519792734210415483908210, 8.012508253808315732787284760582, 9.760703670518298980209544551347, 10.52934633846762820250617412330, 11.22129295774511130759574924498, 12.25513640958955714578749140606, 12.81636463849333674022754518689

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