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Properties

Label 2-538-269.191-c1-0-5
Degree 22
Conductor 538538
Sign 0.957+0.289i0.957 + 0.289i
Analytic cond. 4.295954.29595
Root an. cond. 2.072662.07266
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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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

Λ(s)=(538s/2 ΓC(s) L(s)=((0.957+0.289i) Λ‾(2−s)\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}
Λ(s)=(538s/2 ΓC(s+1/2) L(s)=((0.957+0.289i) Λ‾(1−s)\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: 22
Conductor: 538538    =    2⋅2692 \cdot 269
Sign: 0.957+0.289i0.957 + 0.289i
Analytic conductor: 4.295954.29595
Root analytic conductor: 2.072662.07266
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ538(191,⋅)\chi_{538} (191, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 538, ( :1/2), 0.957+0.289i)(2,\ 538,\ (\ :1/2),\ 0.957 + 0.289i)

Particular Values

L(1)L(1) ≈\approx 0.811589−0.119934i0.811589 - 0.119934i
L(12)L(\frac12) ≈\approx 0.811589−0.119934i0.811589 - 0.119934i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(0.186+0.982i)T 1 + (0.186 + 0.982i)T
269 1+(−15.6−4.98i)T 1 + (-15.6 - 4.98i)T
good3 1+(1.94−0.0455i)T+(2.99−0.140i)T2 1 + (1.94 - 0.0455i)T + (2.99 - 0.140i)T^{2}
5 1+(−0.163+0.619i)T+(−4.34−2.46i)T2 1 + (-0.163 + 0.619i)T + (-4.34 - 2.46i)T^{2}
7 1+(−0.479−1.06i)T+(−4.65+5.23i)T2 1 + (-0.479 - 1.06i)T + (-4.65 + 5.23i)T^{2}
11 1+(1.70+1.91i)T+(−1.28+10.9i)T2 1 + (1.70 + 1.91i)T + (-1.28 + 10.9i)T^{2}
13 1+(0.242−3.44i)T+(−12.8−1.82i)T2 1 + (0.242 - 3.44i)T + (-12.8 - 1.82i)T^{2}
17 1+(−1.22−1.08i)T+(1.98+16.8i)T2 1 + (-1.22 - 1.08i)T + (1.98 + 16.8i)T^{2}
19 1+(−6.19−4.11i)T+(7.37+17.5i)T2 1 + (-6.19 - 4.11i)T + (7.37 + 17.5i)T^{2}
23 1+(0.138+5.89i)T+(−22.9+1.07i)T2 1 + (0.138 + 5.89i)T + (-22.9 + 1.07i)T^{2}
29 1+(−1.20−2.12i)T+(−14.8+24.8i)T2 1 + (-1.20 - 2.12i)T + (-14.8 + 24.8i)T^{2}
31 1+(2.89+0.341i)T+(30.1+7.20i)T2 1 + (2.89 + 0.341i)T + (30.1 + 7.20i)T^{2}
37 1+(−1.51+2.82i)T+(−20.4−30.8i)T2 1 + (-1.51 + 2.82i)T + (-20.4 - 30.8i)T^{2}
41 1+(−11.2−2.12i)T+(38.1+15.0i)T2 1 + (-11.2 - 2.12i)T + (38.1 + 15.0i)T^{2}
43 1+(−7.41+4.20i)T+(22.0−36.8i)T2 1 + (-7.41 + 4.20i)T + (22.0 - 36.8i)T^{2}
47 1+(9.33+2.69i)T+(39.7+25.0i)T2 1 + (9.33 + 2.69i)T + (39.7 + 25.0i)T^{2}
53 1+(−5.62−7.66i)T+(−15.9+50.5i)T2 1 + (-5.62 - 7.66i)T + (-15.9 + 50.5i)T^{2}
59 1+(1.54+3.93i)T+(−43.1+40.2i)T2 1 + (1.54 + 3.93i)T + (-43.1 + 40.2i)T^{2}
61 1+(−6.53−8.08i)T+(−12.7+59.6i)T2 1 + (-6.53 - 8.08i)T + (-12.7 + 59.6i)T^{2}
67 1+(0.0617+0.0243i)T+(49.0+45.6i)T2 1 + (0.0617 + 0.0243i)T + (49.0 + 45.6i)T^{2}
71 1+(7.43−10.6i)T+(−24.4−66.6i)T2 1 + (7.43 - 10.6i)T + (-24.4 - 66.6i)T^{2}
73 1+(4.90−13.3i)T+(−55.6−47.2i)T2 1 + (4.90 - 13.3i)T + (-55.6 - 47.2i)T^{2}
79 1+(−2.97−2.28i)T+(20.1+76.3i)T2 1 + (-2.97 - 2.28i)T + (20.1 + 76.3i)T^{2}
83 1+(−0.981−10.4i)T+(−81.5+15.4i)T2 1 + (-0.981 - 10.4i)T + (-81.5 + 15.4i)T^{2}
89 1+(0.668+4.03i)T+(−84.2+28.6i)T2 1 + (0.668 + 4.03i)T + (-84.2 + 28.6i)T^{2}
97 1+(11.0−13.6i)T+(−20.3−94.8i)T2 1 + (11.0 - 13.6i)T + (-20.3 - 94.8i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

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

Graph of the ZZ-function along the critical line