↑

Properties

Label 2-117-13.10-c3-0-4
Degree $2$
Conductor $117$
Sign $-0.980 - 0.197i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.83 + 1.63i)2-s + (1.37 − 2.38i)4-s + 17.5i·5-s + (23.1 + 13.3i)7-s − 17.2i·8-s + (−28.7 − 49.8i)10-s + (18.5 − 10.7i)11-s + (−8.67 + 46.0i)13-s − 87.5·14-s + (39.2 + 67.9i)16-s + (−41.9 + 72.7i)17-s + (−66.7 − 38.5i)19-s + (41.8 + 24.1i)20-s + (−35.0 + 60.7i)22-s + (−71.0 − 123. i)23-s + ⋯
L(s)  = 1  + (−1.00 + 0.579i)2-s + (0.171 − 0.297i)4-s + 1.56i·5-s + (1.24 + 0.720i)7-s − 0.760i·8-s + (−0.909 − 1.57i)10-s + (0.508 − 0.293i)11-s + (−0.185 + 0.982i)13-s − 1.67·14-s + (0.612 + 1.06i)16-s + (−0.598 + 1.03i)17-s + (−0.806 − 0.465i)19-s + (0.467 + 0.269i)20-s + (−0.340 + 0.588i)22-s + (−0.644 − 1.11i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.197i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.980 - 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.980 - 0.197i$
Analytic conductor: \(6.90322\)
Root analytic conductor: \(2.62739\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :3/2),\ -0.980 - 0.197i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0840101 + 0.841591i\)
\(L(\frac12)\) \(\approx\) \(0.0840101 + 0.841591i\)
\(L(\frac{5}{2})\) 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 \( 1 + (8.67 - 46.0i)T \)
good2 \( 1 + (2.83 - 1.63i)T + (4 - 6.92i)T^{2} \)
5 \( 1 - 17.5iT - 125T^{2} \)
7 \( 1 + (-23.1 - 13.3i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (-18.5 + 10.7i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (41.9 - 72.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (66.7 + 38.5i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (71.0 + 123. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-67.1 - 116. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 122. iT - 2.97e4T^{2} \)
37 \( 1 + (192. - 111. i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-171. + 99.1i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-77.3 + 133. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 78.7iT - 1.03e5T^{2} \)
53 \( 1 - 477.T + 1.48e5T^{2} \)
59 \( 1 + (37.1 + 21.4i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (248. - 430. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-419. + 242. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (331. + 191. i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 193. iT - 3.89e5T^{2} \)
79 \( 1 - 1.04e3T + 4.93e5T^{2} \)
83 \( 1 - 861. iT - 5.71e5T^{2} \)
89 \( 1 + (838. - 483. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-512. - 295. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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

−13.93661137604779442992507578518, −12.25988011178112923762990800215, −11.13376204917481685287342788630, −10.41060459289378537133219162432, −8.973889755755620044427793058210, −8.247302019029672459705539403364, −6.99098009863400932907583894997, −6.24883226403693915474549877793, −4.12655750778216299118120265220, −2.15213172624211459106428519996, 0.65392120080460881131308576823, 1.76382270438209976503404031104, 4.43177943267779533563659555994, 5.37169206346585805094194982882, 7.66265684476328590154767902106, 8.417190671189856879784733006047, 9.321433830510295253165126757276, 10.34878473091654171323287088246, 11.42799239189184570775296472809, 12.25801672670270880403300870579

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