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Properties

Label 2-538-269.191-c1-0-1
Degree 22
Conductor 538538
Sign −0.996+0.0877i-0.996 + 0.0877i
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.186 + 0.982i)2-s + (0.0181 − 0.000425i)3-s + (−0.930 + 0.366i)4-s + (−0.713 + 2.70i)5-s + (0.00380 + 0.0177i)6-s + (1.80 + 4.02i)7-s + (−0.533 − 0.845i)8-s + (−2.99 + 0.140i)9-s + (−2.79 − 0.196i)10-s + (−2.74 − 3.08i)11-s + (−0.0167 + 0.00704i)12-s + (0.108 − 1.54i)13-s + (−3.61 + 2.52i)14-s + (−0.0118 + 0.0494i)15-s + (0.731 − 0.681i)16-s + (−2.87 − 2.55i)17-s + ⋯
L(s)  = 1  + (0.131 + 0.694i)2-s + (0.0104 − 0.000245i)3-s + (−0.465 + 0.183i)4-s + (−0.319 + 1.21i)5-s + (0.00155 + 0.00724i)6-s + (0.682 + 1.51i)7-s + (−0.188 − 0.299i)8-s + (−0.998 + 0.0468i)9-s + (−0.882 − 0.0622i)10-s + (−0.827 − 0.930i)11-s + (−0.00483 + 0.00203i)12-s + (0.0301 − 0.428i)13-s + (−0.965 + 0.674i)14-s + (−0.00304 + 0.0127i)15-s + (0.182 − 0.170i)16-s + (−0.697 − 0.620i)17-s + ⋯

Functional equation

Λ(s)=(538s/2 ΓC(s) L(s)=((−0.996+0.0877i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0877i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(538s/2 ΓC(s+1/2) L(s)=((−0.996+0.0877i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0877i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 538538    =    2⋅2692 \cdot 269
Sign: −0.996+0.0877i-0.996 + 0.0877i
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), −0.996+0.0877i)(2,\ 538,\ (\ :1/2),\ -0.996 + 0.0877i)

Particular Values

L(1)L(1) ≈\approx 0.0425125−0.967594i0.0425125 - 0.967594i
L(12)L(\frac12) ≈\approx 0.0425125−0.967594i0.0425125 - 0.967594i
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+(−16.3+1.18i)T 1 + (-16.3 + 1.18i)T
good3 1+(−0.0181+0.000425i)T+(2.99−0.140i)T2 1 + (-0.0181 + 0.000425i)T + (2.99 - 0.140i)T^{2}
5 1+(0.713−2.70i)T+(−4.34−2.46i)T2 1 + (0.713 - 2.70i)T + (-4.34 - 2.46i)T^{2}
7 1+(−1.80−4.02i)T+(−4.65+5.23i)T2 1 + (-1.80 - 4.02i)T + (-4.65 + 5.23i)T^{2}
11 1+(2.74+3.08i)T+(−1.28+10.9i)T2 1 + (2.74 + 3.08i)T + (-1.28 + 10.9i)T^{2}
13 1+(−0.108+1.54i)T+(−12.8−1.82i)T2 1 + (-0.108 + 1.54i)T + (-12.8 - 1.82i)T^{2}
17 1+(2.87+2.55i)T+(1.98+16.8i)T2 1 + (2.87 + 2.55i)T + (1.98 + 16.8i)T^{2}
19 1+(−4.07−2.70i)T+(7.37+17.5i)T2 1 + (-4.07 - 2.70i)T + (7.37 + 17.5i)T^{2}
23 1+(−0.0466−1.98i)T+(−22.9+1.07i)T2 1 + (-0.0466 - 1.98i)T + (-22.9 + 1.07i)T^{2}
29 1+(−1.10−1.94i)T+(−14.8+24.8i)T2 1 + (-1.10 - 1.94i)T + (-14.8 + 24.8i)T^{2}
31 1+(1.43+0.169i)T+(30.1+7.20i)T2 1 + (1.43 + 0.169i)T + (30.1 + 7.20i)T^{2}
37 1+(0.326−0.608i)T+(−20.4−30.8i)T2 1 + (0.326 - 0.608i)T + (-20.4 - 30.8i)T^{2}
41 1+(−0.468−0.0888i)T+(38.1+15.0i)T2 1 + (-0.468 - 0.0888i)T + (38.1 + 15.0i)T^{2}
43 1+(10.4−5.91i)T+(22.0−36.8i)T2 1 + (10.4 - 5.91i)T + (22.0 - 36.8i)T^{2}
47 1+(5.51+1.59i)T+(39.7+25.0i)T2 1 + (5.51 + 1.59i)T + (39.7 + 25.0i)T^{2}
53 1+(−5.74−7.83i)T+(−15.9+50.5i)T2 1 + (-5.74 - 7.83i)T + (-15.9 + 50.5i)T^{2}
59 1+(−0.566−1.43i)T+(−43.1+40.2i)T2 1 + (-0.566 - 1.43i)T + (-43.1 + 40.2i)T^{2}
61 1+(−5.45−6.74i)T+(−12.7+59.6i)T2 1 + (-5.45 - 6.74i)T + (-12.7 + 59.6i)T^{2}
67 1+(−6.90−2.71i)T+(49.0+45.6i)T2 1 + (-6.90 - 2.71i)T + (49.0 + 45.6i)T^{2}
71 1+(−6.08+8.71i)T+(−24.4−66.6i)T2 1 + (-6.08 + 8.71i)T + (-24.4 - 66.6i)T^{2}
73 1+(4.77−13.0i)T+(−55.6−47.2i)T2 1 + (4.77 - 13.0i)T + (-55.6 - 47.2i)T^{2}
79 1+(−11.9−9.20i)T+(20.1+76.3i)T2 1 + (-11.9 - 9.20i)T + (20.1 + 76.3i)T^{2}
83 1+(1.39+14.8i)T+(−81.5+15.4i)T2 1 + (1.39 + 14.8i)T + (-81.5 + 15.4i)T^{2}
89 1+(−2.40−14.5i)T+(−84.2+28.6i)T2 1 + (-2.40 - 14.5i)T + (-84.2 + 28.6i)T^{2}
97 1+(−3.27+4.04i)T+(−20.3−94.8i)T2 1 + (-3.27 + 4.04i)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

−11.46203560939207826137621843106, −10.52918562469548305278143190260, −9.251883960891406824034743617696, −8.359668381767209544673956810016, −7.85061850593693538016664250999, −6.65853028171629013113739527261, −5.64145707883593347802543709203, −5.20468156872112387263659729982, −3.28449015692008147318777104657, −2.62054989245534308240101089336, 0.52757378545028359423182493846, 1.97975594316244933735286570183, 3.66271809894041066304055031367, 4.69071535920127714392281639716, 5.13036212060368556714482396726, 6.83706023575108433912086836680, 7.956033898110376881215044372098, 8.554198248201803942383676705017, 9.600689333272792828008747264843, 10.50791160211931857320254780073

Graph of the ZZ-function along the critical line