Properties

Label 2-538-269.100-c1-0-9
Degree $2$
Conductor $538$
Sign $0.957 - 0.289i$
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 + (−1.94 − 0.0455i)3-s + (−0.930 − 0.366i)4-s + (0.163 + 0.619i)5-s + (0.406 − 1.89i)6-s + (0.479 − 1.06i)7-s + (0.533 − 0.845i)8-s + (0.773 + 0.0363i)9-s + (−0.638 + 0.0449i)10-s + (−1.70 + 1.91i)11-s + (1.79 + 0.754i)12-s + (−0.242 − 3.44i)13-s + (0.959 + 0.669i)14-s + (−0.288 − 1.20i)15-s + (0.731 + 0.681i)16-s + (1.22 − 1.08i)17-s + ⋯
L(s)  = 1  + (−0.131 + 0.694i)2-s + (−1.12 − 0.0262i)3-s + (−0.465 − 0.183i)4-s + (0.0730 + 0.276i)5-s + (0.166 − 0.775i)6-s + (0.181 − 0.403i)7-s + (0.188 − 0.299i)8-s + (0.257 + 0.0121i)9-s + (−0.201 + 0.0142i)10-s + (−0.512 + 0.576i)11-s + (0.516 + 0.217i)12-s + (−0.0673 − 0.955i)13-s + (0.256 + 0.178i)14-s + (−0.0746 − 0.312i)15-s + (0.182 + 0.170i)16-s + (0.296 − 0.263i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(538\)    =    \(2 \cdot 269\)
Sign: $0.957 - 0.289i$
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: \((2,\ 538,\ (\ :1/2),\ 0.957 - 0.289i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.811589 + 0.119934i\)
\(L(\frac12)\) \(\approx\) \(0.811589 + 0.119934i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-15.6 + 4.98i)T \)
good3 \( 1 + (1.94 + 0.0455i)T + (2.99 + 0.140i)T^{2} \)
5 \( 1 + (-0.163 - 0.619i)T + (-4.34 + 2.46i)T^{2} \)
7 \( 1 + (-0.479 + 1.06i)T + (-4.65 - 5.23i)T^{2} \)
11 \( 1 + (1.70 - 1.91i)T + (-1.28 - 10.9i)T^{2} \)
13 \( 1 + (0.242 + 3.44i)T + (-12.8 + 1.82i)T^{2} \)
17 \( 1 + (-1.22 + 1.08i)T + (1.98 - 16.8i)T^{2} \)
19 \( 1 + (-6.19 + 4.11i)T + (7.37 - 17.5i)T^{2} \)
23 \( 1 + (0.138 - 5.89i)T + (-22.9 - 1.07i)T^{2} \)
29 \( 1 + (-1.20 + 2.12i)T + (-14.8 - 24.8i)T^{2} \)
31 \( 1 + (2.89 - 0.341i)T + (30.1 - 7.20i)T^{2} \)
37 \( 1 + (-1.51 - 2.82i)T + (-20.4 + 30.8i)T^{2} \)
41 \( 1 + (-11.2 + 2.12i)T + (38.1 - 15.0i)T^{2} \)
43 \( 1 + (-7.41 - 4.20i)T + (22.0 + 36.8i)T^{2} \)
47 \( 1 + (9.33 - 2.69i)T + (39.7 - 25.0i)T^{2} \)
53 \( 1 + (-5.62 + 7.66i)T + (-15.9 - 50.5i)T^{2} \)
59 \( 1 + (1.54 - 3.93i)T + (-43.1 - 40.2i)T^{2} \)
61 \( 1 + (-6.53 + 8.08i)T + (-12.7 - 59.6i)T^{2} \)
67 \( 1 + (0.0617 - 0.0243i)T + (49.0 - 45.6i)T^{2} \)
71 \( 1 + (7.43 + 10.6i)T + (-24.4 + 66.6i)T^{2} \)
73 \( 1 + (4.90 + 13.3i)T + (-55.6 + 47.2i)T^{2} \)
79 \( 1 + (-2.97 + 2.28i)T + (20.1 - 76.3i)T^{2} \)
83 \( 1 + (-0.981 + 10.4i)T + (-81.5 - 15.4i)T^{2} \)
89 \( 1 + (0.668 - 4.03i)T + (-84.2 - 28.6i)T^{2} \)
97 \( 1 + (11.0 + 13.6i)T + (-20.3 + 94.8i)T^{2} \)
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

−10.85014649044262985674598691880, −10.05706286967337456800362296853, −9.191232079546174985222403106811, −7.75513477546352024365714986509, −7.32566097307051639002132954641, −6.19900095753597436406807932682, −5.37147062608637782569266497739, −4.70536547554852999946689519262, −3.02826988913102483369346772415, −0.77331662405294231027607731925, 1.04528109672278420231427349826, 2.68159063899332972316948647672, 4.13429556675201460568855886441, 5.29612287501830719930034622827, 5.82114439277700761041136410330, 7.11808393723636212379943792443, 8.337632388386314407588475052079, 9.123920947763277987584316737342, 10.14967981370806264630448985955, 10.94470526808441439236302295445

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