Properties

Degree 2
Conductor $ 2^{2} $
Sign $-0.718 + 0.695i$
Motivic weight 36
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (9.83e4 + 2.42e5i)2-s + 4.79e8i·3-s + (−4.93e10 + 4.78e10i)4-s + 6.60e12·5-s + (−1.16e14 + 4.71e13i)6-s − 2.27e14i·7-s + (−1.64e16 − 7.28e15i)8-s − 7.98e16·9-s + (6.49e17 + 1.60e18i)10-s + 8.91e18i·11-s + (−2.29e19 − 2.36e19i)12-s − 1.69e20·13-s + (5.51e19 − 2.23e19i)14-s + 3.16e21i·15-s + (1.50e20 − 4.71e21i)16-s − 1.04e22·17-s + ⋯
L(s)  = 1  + (0.375 + 0.926i)2-s + 1.23i·3-s + (−0.718 + 0.695i)4-s + 1.73·5-s + (−1.14 + 0.464i)6-s − 0.139i·7-s + (−0.914 − 0.404i)8-s − 0.531·9-s + (0.649 + 1.60i)10-s + 1.60i·11-s + (−0.861 − 0.888i)12-s − 1.50·13-s + (0.129 − 0.0523i)14-s + 2.14i·15-s + (0.0318 − 0.999i)16-s − 0.741·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.718 + 0.695i)\, \overline{\Lambda}(37-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+18) \, L(s)\cr =\mathstrut & (-0.718 + 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4\)    =    \(2^{2}\)
\( \varepsilon \)  =  $-0.718 + 0.695i$
motivic weight  =  \(36\)
character  :  $\chi_{4} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 4,\ (\ :18),\ -0.718 + 0.695i)\)
\(L(\frac{37}{2})\)  \(\approx\)  \(2.304150998\)
\(L(\frac12)\)  \(\approx\)  \(2.304150998\)
\(L(19)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 2$,\(F_p(T)\) is a polynomial of degree 2. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (-9.83e4 - 2.42e5i)T \)
good3 \( 1 - 4.79e8iT - 1.50e17T^{2} \)
5 \( 1 - 6.60e12T + 1.45e25T^{2} \)
7 \( 1 + 2.27e14iT - 2.65e30T^{2} \)
11 \( 1 - 8.91e18iT - 3.09e37T^{2} \)
13 \( 1 + 1.69e20T + 1.26e40T^{2} \)
17 \( 1 + 1.04e22T + 1.97e44T^{2} \)
19 \( 1 + 7.97e21iT - 1.08e46T^{2} \)
23 \( 1 - 1.29e24iT - 1.05e49T^{2} \)
29 \( 1 - 6.13e25T + 4.42e52T^{2} \)
31 \( 1 + 7.63e26iT - 4.88e53T^{2} \)
37 \( 1 + 1.94e28T + 2.85e56T^{2} \)
41 \( 1 - 5.45e28T + 1.14e58T^{2} \)
43 \( 1 - 1.65e29iT - 6.38e58T^{2} \)
47 \( 1 - 5.00e29iT - 1.56e60T^{2} \)
53 \( 1 + 2.38e30T + 1.18e62T^{2} \)
59 \( 1 + 7.57e31iT - 5.63e63T^{2} \)
61 \( 1 - 1.15e32T + 1.87e64T^{2} \)
67 \( 1 - 9.05e32iT - 5.47e65T^{2} \)
71 \( 1 - 7.34e32iT - 4.41e66T^{2} \)
73 \( 1 - 3.68e33T + 1.20e67T^{2} \)
79 \( 1 - 2.27e34iT - 2.06e68T^{2} \)
83 \( 1 + 1.22e34iT - 1.22e69T^{2} \)
89 \( 1 - 8.48e32T + 1.50e70T^{2} \)
97 \( 1 + 2.86e35T + 3.34e71T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−17.02654698387953010361322678068, −15.35423063071234658003325095716, −14.32574639326385638325478331917, −12.80262774362269004083737022184, −9.963165282627638364112526421472, −9.408167164229154411184240414001, −6.97894467880693864368655179296, −5.31785067991563072237288560997, −4.44220464997941431474663410959, −2.33038252803916961404326009595, 0.58027044328429848954052299218, 1.82228881647261449157466752290, 2.69466303464028805680902783681, 5.31333223750731999286280776542, 6.49338979790982304016466413644, 8.907129214494243769376636809189, 10.43202071604457491758292444975, 12.26634362448055478141744891888, 13.43480109205364294385175319291, 14.15696934034636227978694986311

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