↑

Properties

Label 2-138-69.68-c5-0-12
Degree $2$
Conductor $138$
Sign $0.961 + 0.275i$
Analytic cond. $22.1329$
Root an. cond. $4.70456$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4i·2-s + (−11.7 − 10.2i)3-s − 16·4-s − 107.·5-s + (40.9 − 46.9i)6-s + 192. i·7-s − 64i·8-s + (33.0 + 240. i)9-s − 429. i·10-s − 525.·11-s + (187. + 163. i)12-s − 1.09e3·13-s − 769.·14-s + (1.26e3 + 1.10e3i)15-s + 256·16-s + 665.·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.753 − 0.657i)3-s − 0.5·4-s − 1.92·5-s + (0.464 − 0.532i)6-s + 1.48i·7-s − 0.353i·8-s + (0.135 + 0.990i)9-s − 1.35i·10-s − 1.31·11-s + (0.376 + 0.328i)12-s − 1.80·13-s − 1.04·14-s + (1.44 + 1.26i)15-s + 0.250·16-s + 0.558·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.275i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.961 + 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138\)    =    \(2 \cdot 3 \cdot 23\)
Sign: $0.961 + 0.275i$
Analytic conductor: \(22.1329\)
Root analytic conductor: \(4.70456\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{138} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 138,\ (\ :5/2),\ 0.961 + 0.275i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.1266281631\)
\(L(\frac12)\) \(\approx\) \(0.1266281631\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 4iT \)
3 \( 1 + (11.7 + 10.2i)T \)
23 \( 1 + (1.07e3 + 2.29e3i)T \)
good5 \( 1 + 107.T + 3.12e3T^{2} \)
7 \( 1 - 192. iT - 1.68e4T^{2} \)
11 \( 1 + 525.T + 1.61e5T^{2} \)
13 \( 1 + 1.09e3T + 3.71e5T^{2} \)
17 \( 1 - 665.T + 1.41e6T^{2} \)
19 \( 1 - 365. iT - 2.47e6T^{2} \)
29 \( 1 - 2.16e3iT - 2.05e7T^{2} \)
31 \( 1 + 4.02e3T + 2.86e7T^{2} \)
37 \( 1 - 1.17e4iT - 6.93e7T^{2} \)
41 \( 1 + 1.01e4iT - 1.15e8T^{2} \)
43 \( 1 - 3.67e3iT - 1.47e8T^{2} \)
47 \( 1 + 8.07e3iT - 2.29e8T^{2} \)
53 \( 1 + 1.21e4T + 4.18e8T^{2} \)
59 \( 1 - 3.84e3iT - 7.14e8T^{2} \)
61 \( 1 - 2.07e4iT - 8.44e8T^{2} \)
67 \( 1 - 2.01e4iT - 1.35e9T^{2} \)
71 \( 1 - 5.75e4iT - 1.80e9T^{2} \)
73 \( 1 + 3.44e4T + 2.07e9T^{2} \)
79 \( 1 + 1.32e4iT - 3.07e9T^{2} \)
83 \( 1 + 7.45e3T + 3.93e9T^{2} \)
89 \( 1 + 1.89e4T + 5.58e9T^{2} \)
97 \( 1 - 4.13e4iT - 8.58e9T^{2} \)
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

−12.31392107270471139446229171069, −11.60424385807655410806796205196, −10.27403559507651136418102677092, −8.521926787529657992513344170354, −7.79618102985323477916785890973, −7.04562585381019605295525206589, −5.49011217584484283560427482422, −4.72291729006559250357605209793, −2.70822820015658941702815076399, −0.11864740370128781950151705516, 0.46188220572768892223732827252, 3.27869489703268300573698018197, 4.25562745112200655340535288396, 5.04477932907404319737733285098, 7.33392573200076607019268155351, 7.78928083238186439028055804978, 9.635685105704963964104620163396, 10.55828652457778538917987763118, 11.18627253706708526695871696443, 12.10858531120024988327093666167

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