↑

Properties

Label 2-538-269.100-c1-0-5
Degree $2$
Conductor $538$
Sign $-0.904 - 0.427i$
Analytic cond. $4.29595$
Root an. cond. $2.07266$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.186 + 0.982i)2-s + (0.411 + 0.00964i)3-s + (−0.930 − 0.366i)4-s + (0.415 + 1.57i)5-s + (−0.0862 + 0.402i)6-s + (−0.208 + 0.464i)7-s + (0.533 − 0.845i)8-s + (−2.82 − 0.132i)9-s + (−1.62 + 0.114i)10-s + (−0.181 + 0.203i)11-s + (−0.379 − 0.159i)12-s + (0.295 + 4.19i)13-s + (−0.417 − 0.291i)14-s + (0.155 + 0.652i)15-s + (0.731 + 0.681i)16-s + (−2.44 + 2.17i)17-s + ⋯
L(s)  = 1  + (−0.131 + 0.694i)2-s + (0.237 + 0.00557i)3-s + (−0.465 − 0.183i)4-s + (0.185 + 0.704i)5-s + (−0.0351 + 0.164i)6-s + (−0.0787 + 0.175i)7-s + (0.188 − 0.299i)8-s + (−0.942 − 0.0442i)9-s + (−0.514 + 0.0362i)10-s + (−0.0546 + 0.0614i)11-s + (−0.109 − 0.0461i)12-s + (0.0819 + 1.16i)13-s + (−0.111 − 0.0778i)14-s + (0.0402 + 0.168i)15-s + (0.182 + 0.170i)16-s + (−0.593 + 0.527i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 - 0.427i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.904 - 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(538\)    =    \(2 \cdot 269\)
Sign: $-0.904 - 0.427i$
Analytic conductor: \(4.29595\)
Root analytic conductor: \(2.07266\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{538} (369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 538,\ (\ :1/2),\ -0.904 - 0.427i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.217824 + 0.971037i\)
\(L(\frac12)\) \(\approx\) \(0.217824 + 0.971037i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.186 - 0.982i)T \)
269 \( 1 + (14.9 + 6.77i)T \)
good3 \( 1 + (-0.411 - 0.00964i)T + (2.99 + 0.140i)T^{2} \)
5 \( 1 + (-0.415 - 1.57i)T + (-4.34 + 2.46i)T^{2} \)
7 \( 1 + (0.208 - 0.464i)T + (-4.65 - 5.23i)T^{2} \)
11 \( 1 + (0.181 - 0.203i)T + (-1.28 - 10.9i)T^{2} \)
13 \( 1 + (-0.295 - 4.19i)T + (-12.8 + 1.82i)T^{2} \)
17 \( 1 + (2.44 - 2.17i)T + (1.98 - 16.8i)T^{2} \)
19 \( 1 + (2.59 - 1.72i)T + (7.37 - 17.5i)T^{2} \)
23 \( 1 + (0.141 - 6.02i)T + (-22.9 - 1.07i)T^{2} \)
29 \( 1 + (-3.87 + 6.83i)T + (-14.8 - 24.8i)T^{2} \)
31 \( 1 + (4.64 - 0.547i)T + (30.1 - 7.20i)T^{2} \)
37 \( 1 + (-3.47 - 6.47i)T + (-20.4 + 30.8i)T^{2} \)
41 \( 1 + (2.28 - 0.434i)T + (38.1 - 15.0i)T^{2} \)
43 \( 1 + (3.06 + 1.74i)T + (22.0 + 36.8i)T^{2} \)
47 \( 1 + (3.54 - 1.02i)T + (39.7 - 25.0i)T^{2} \)
53 \( 1 + (2.96 - 4.04i)T + (-15.9 - 50.5i)T^{2} \)
59 \( 1 + (-4.59 + 11.6i)T + (-43.1 - 40.2i)T^{2} \)
61 \( 1 + (6.85 - 8.47i)T + (-12.7 - 59.6i)T^{2} \)
67 \( 1 + (-6.35 + 2.50i)T + (49.0 - 45.6i)T^{2} \)
71 \( 1 + (-3.21 - 4.60i)T + (-24.4 + 66.6i)T^{2} \)
73 \( 1 + (-0.0380 - 0.103i)T + (-55.6 + 47.2i)T^{2} \)
79 \( 1 + (-11.4 + 8.83i)T + (20.1 - 76.3i)T^{2} \)
83 \( 1 + (-0.934 + 9.94i)T + (-81.5 - 15.4i)T^{2} \)
89 \( 1 + (-2.53 + 15.2i)T + (-84.2 - 28.6i)T^{2} \)
97 \( 1 + (-6.09 - 7.53i)T + (-20.3 + 94.8i)T^{2} \)
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

−11.17586498200469795656764898161, −10.18326365222849336576306552073, −9.255779209490577090170533746453, −8.536520985806146910612140746538, −7.62602485829648389048484438256, −6.49459253890644687373749002685, −6.04718611496171052842888726353, −4.69519590167467681597091296977, −3.47140303092021573753283060279, −2.11637069879361549567020338768, 0.56903264891795374606393689674, 2.34315022262253750290728494954, 3.36261983895167198832315924409, 4.73007521657750157799894401335, 5.52916907296430524483392026978, 6.84763157147800633758048687282, 8.207039111721460865732260308444, 8.698422045191316289539922249008, 9.487355071993917201366428699410, 10.65442932319714107479884890314

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