↑

Properties

Label 2-38-19.18-c8-0-4
Degree $2$
Conductor $38$
Sign $0.690 - 0.722i$
Analytic cond. $15.4803$
Root an. cond. $3.93451$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 11.3i·2-s + 76.0i·3-s − 128.·4-s + 1.15e3·5-s + 859.·6-s − 2.86e3·7-s + 1.44e3i·8-s + 784.·9-s − 1.30e4i·10-s − 4.90e3·11-s − 9.72e3i·12-s + 4.16e4i·13-s + 3.24e4i·14-s + 8.78e4i·15-s + 1.63e4·16-s + 5.13e4·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.938i·3-s − 0.500·4-s + 1.84·5-s + 0.663·6-s − 1.19·7-s + 0.353i·8-s + 0.119·9-s − 1.30i·10-s − 0.335·11-s − 0.469i·12-s + 1.45i·13-s + 0.845i·14-s + 1.73i·15-s + 0.250·16-s + 0.614·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.690 - 0.722i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.690 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.690 - 0.722i$
Analytic conductor: \(15.4803\)
Root analytic conductor: \(3.93451\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :4),\ 0.690 - 0.722i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.88999 + 0.808028i\)
\(L(\frac12)\) \(\approx\) \(1.88999 + 0.808028i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 11.3iT \)
19 \( 1 + (-9.42e4 - 9.00e4i)T \)
good3 \( 1 - 76.0iT - 6.56e3T^{2} \)
5 \( 1 - 1.15e3T + 3.90e5T^{2} \)
7 \( 1 + 2.86e3T + 5.76e6T^{2} \)
11 \( 1 + 4.90e3T + 2.14e8T^{2} \)
13 \( 1 - 4.16e4iT - 8.15e8T^{2} \)
17 \( 1 - 5.13e4T + 6.97e9T^{2} \)
23 \( 1 + 8.85e4T + 7.83e10T^{2} \)
29 \( 1 - 1.80e5iT - 5.00e11T^{2} \)
31 \( 1 - 8.88e5iT - 8.52e11T^{2} \)
37 \( 1 - 2.53e6iT - 3.51e12T^{2} \)
41 \( 1 + 3.37e6iT - 7.98e12T^{2} \)
43 \( 1 - 3.46e6T + 1.16e13T^{2} \)
47 \( 1 + 5.32e6T + 2.38e13T^{2} \)
53 \( 1 + 7.74e6iT - 6.22e13T^{2} \)
59 \( 1 + 1.04e7iT - 1.46e14T^{2} \)
61 \( 1 + 1.38e7T + 1.91e14T^{2} \)
67 \( 1 + 1.81e7iT - 4.06e14T^{2} \)
71 \( 1 + 1.83e7iT - 6.45e14T^{2} \)
73 \( 1 - 4.49e7T + 8.06e14T^{2} \)
79 \( 1 + 5.30e7iT - 1.51e15T^{2} \)
83 \( 1 - 1.92e7T + 2.25e15T^{2} \)
89 \( 1 - 2.43e7iT - 3.93e15T^{2} \)
97 \( 1 - 7.02e7iT - 7.83e15T^{2} \)
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

−14.33892642941473638154487562015, −13.52046120836136089581557937073, −12.37947245608911310183906770672, −10.52446290510328273772943167264, −9.749814956133557745312294084192, −9.256112567339386318213742324984, −6.47231218663483246076229510294, −5.08250722121081099599792856752, −3.33121663415243292146495306168, −1.70361202747989007264294279948, 0.866238499730233161697991663889, 2.69084093703879231652152376279, 5.56314421449142018519631607578, 6.34351081101101784849711023065, 7.61813252197831345575273888156, 9.431651912262878771016516852353, 10.19844663009972791568415521731, 12.80422300845637026008834177983, 13.10595325713556282412099348333, 14.08537024339868829588477662525

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