Properties

Label 2-2e2-4.3-c38-0-10
Degree $2$
Conductor $4$
Sign $0.875 - 0.483i$
Analytic cond. $36.5853$
Root an. cond. $6.04858$
Motivic weight $38$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.30e5 + 5.07e5i)2-s + 1.21e9i·3-s + (−2.40e11 + 1.32e11i)4-s − 7.30e12·5-s + (−6.16e14 + 1.58e14i)6-s − 7.26e15i·7-s + (−9.89e16 − 1.04e17i)8-s − 1.25e17·9-s + (−9.55e17 − 3.71e18i)10-s − 4.95e19i·11-s + (−1.61e20 − 2.92e20i)12-s + 1.33e21·13-s + (3.68e21 − 9.50e20i)14-s − 8.87e21i·15-s + (4.02e22 − 6.39e22i)16-s − 1.14e22·17-s + ⋯
L(s)  = 1  + (0.249 + 0.968i)2-s + 1.04i·3-s + (−0.875 + 0.483i)4-s − 0.383·5-s + (−1.01 + 0.260i)6-s − 0.637i·7-s + (−0.686 − 0.727i)8-s − 0.0931·9-s + (−0.0955 − 0.371i)10-s − 0.809i·11-s + (−0.505 − 0.915i)12-s + 0.912·13-s + (0.617 − 0.159i)14-s − 0.400i·15-s + (0.533 − 0.846i)16-s − 0.0479·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 - 0.483i)\, \overline{\Lambda}(39-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+19) \, L(s)\cr =\mathstrut & (0.875 - 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $0.875 - 0.483i$
Analytic conductor: \(36.5853\)
Root analytic conductor: \(6.04858\)
Motivic weight: \(38\)
Rational: no
Arithmetic: yes
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :19),\ 0.875 - 0.483i)\)

Particular Values

\(L(\frac{39}{2})\) \(\approx\) \(1.462473075\)
\(L(\frac12)\) \(\approx\) \(1.462473075\)
\(L(20)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.30e5 - 5.07e5i)T \)
good3 \( 1 - 1.21e9iT - 1.35e18T^{2} \)
5 \( 1 + 7.30e12T + 3.63e26T^{2} \)
7 \( 1 + 7.26e15iT - 1.29e32T^{2} \)
11 \( 1 + 4.95e19iT - 3.74e39T^{2} \)
13 \( 1 - 1.33e21T + 2.13e42T^{2} \)
17 \( 1 + 1.14e22T + 5.71e46T^{2} \)
19 \( 1 + 5.70e22iT - 3.91e48T^{2} \)
23 \( 1 + 1.17e26iT - 5.56e51T^{2} \)
29 \( 1 + 1.11e28T + 3.72e55T^{2} \)
31 \( 1 + 1.05e28iT - 4.69e56T^{2} \)
37 \( 1 - 9.56e29T + 3.90e59T^{2} \)
41 \( 1 - 5.59e30T + 1.93e61T^{2} \)
43 \( 1 - 7.10e30iT - 1.17e62T^{2} \)
47 \( 1 + 1.17e31iT - 3.46e63T^{2} \)
53 \( 1 + 3.89e32T + 3.33e65T^{2} \)
59 \( 1 + 6.64e33iT - 1.96e67T^{2} \)
61 \( 1 + 6.57e33T + 6.95e67T^{2} \)
67 \( 1 + 3.38e33iT - 2.45e69T^{2} \)
71 \( 1 + 4.35e34iT - 2.22e70T^{2} \)
73 \( 1 - 2.83e35T + 6.40e70T^{2} \)
79 \( 1 + 2.18e36iT - 1.28e72T^{2} \)
83 \( 1 + 1.82e36iT - 8.41e72T^{2} \)
89 \( 1 - 2.05e37T + 1.19e74T^{2} \)
97 \( 1 + 5.02e37T + 3.14e75T^{2} \)
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

−15.94574192417722471810993190353, −14.66480897412232453906226888362, −13.18141565984418723712424620126, −10.94828899540336009908666068675, −9.308472416941573014159133387580, −7.83006963613603868713628590678, −6.07454458118705866801140549020, −4.42529759326610834912145530081, −3.58313572563675500786344049507, −0.45830015107994802837902128881, 1.18000601727445975067507563490, 2.20695388002916384170754432285, 3.91617282357632359746615381138, 5.78719611506204001866744606083, 7.68347830688879028152221477709, 9.395490328421142045034838728254, 11.33640055080473521579559673617, 12.46858779424667515916310172220, 13.51068684936158193404788324746, 15.28968273664610955970464946097

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