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Properties

Label 2-538-269.191-c1-0-13
Degree 22
Conductor 538538
Sign 0.837+0.546i0.837 + 0.546i
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 + (2.60 − 0.0610i)3-s + (−0.930 + 0.366i)4-s + (0.0581 − 0.220i)5-s + (−0.545 − 2.54i)6-s + (2.03 + 4.54i)7-s + (0.533 + 0.845i)8-s + (3.77 − 0.177i)9-s + (−0.227 − 0.0160i)10-s + (−3.64 − 4.09i)11-s + (−2.40 + 1.01i)12-s + (0.357 − 5.07i)13-s + (4.08 − 2.85i)14-s + (0.137 − 0.577i)15-s + (0.731 − 0.681i)16-s + (5.65 + 5.02i)17-s + ⋯
L(s)  = 1  + (−0.131 − 0.694i)2-s + (1.50 − 0.0352i)3-s + (−0.465 + 0.183i)4-s + (0.0259 − 0.0985i)5-s + (−0.222 − 1.03i)6-s + (0.770 + 1.71i)7-s + (0.188 + 0.299i)8-s + (1.25 − 0.0590i)9-s + (−0.0719 − 0.00506i)10-s + (−1.09 − 1.23i)11-s + (−0.692 + 0.291i)12-s + (0.0990 − 1.40i)13-s + (1.09 − 0.761i)14-s + (0.0356 − 0.149i)15-s + (0.182 − 0.170i)16-s + (1.37 + 1.21i)17-s + ⋯

Functional equation

Λ(s)=(538s/2 ΓC(s) L(s)=((0.837+0.546i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 + 0.546i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(538s/2 ΓC(s+1/2) L(s)=((0.837+0.546i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.837 + 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 538538    =    2⋅2692 \cdot 269
Sign: 0.837+0.546i0.837 + 0.546i
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), 0.837+0.546i)(2,\ 538,\ (\ :1/2),\ 0.837 + 0.546i)

Particular Values

L(1)L(1) ≈\approx 2.12664−0.632675i2.12664 - 0.632675i
L(12)L(\frac12) ≈\approx 2.12664−0.632675i2.12664 - 0.632675i
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+(−13.5−9.17i)T 1 + (-13.5 - 9.17i)T
good3 1+(−2.60+0.0610i)T+(2.99−0.140i)T2 1 + (-2.60 + 0.0610i)T + (2.99 - 0.140i)T^{2}
5 1+(−0.0581+0.220i)T+(−4.34−2.46i)T2 1 + (-0.0581 + 0.220i)T + (-4.34 - 2.46i)T^{2}
7 1+(−2.03−4.54i)T+(−4.65+5.23i)T2 1 + (-2.03 - 4.54i)T + (-4.65 + 5.23i)T^{2}
11 1+(3.64+4.09i)T+(−1.28+10.9i)T2 1 + (3.64 + 4.09i)T + (-1.28 + 10.9i)T^{2}
13 1+(−0.357+5.07i)T+(−12.8−1.82i)T2 1 + (-0.357 + 5.07i)T + (-12.8 - 1.82i)T^{2}
17 1+(−5.65−5.02i)T+(1.98+16.8i)T2 1 + (-5.65 - 5.02i)T + (1.98 + 16.8i)T^{2}
19 1+(−2.17−1.44i)T+(7.37+17.5i)T2 1 + (-2.17 - 1.44i)T + (7.37 + 17.5i)T^{2}
23 1+(0.0595+2.54i)T+(−22.9+1.07i)T2 1 + (0.0595 + 2.54i)T + (-22.9 + 1.07i)T^{2}
29 1+(1.19+2.11i)T+(−14.8+24.8i)T2 1 + (1.19 + 2.11i)T + (-14.8 + 24.8i)T^{2}
31 1+(−3.77−0.445i)T+(30.1+7.20i)T2 1 + (-3.77 - 0.445i)T + (30.1 + 7.20i)T^{2}
37 1+(3.66−6.83i)T+(−20.4−30.8i)T2 1 + (3.66 - 6.83i)T + (-20.4 - 30.8i)T^{2}
41 1+(6.87+1.30i)T+(38.1+15.0i)T2 1 + (6.87 + 1.30i)T + (38.1 + 15.0i)T^{2}
43 1+(−2.98+1.69i)T+(22.0−36.8i)T2 1 + (-2.98 + 1.69i)T + (22.0 - 36.8i)T^{2}
47 1+(9.52+2.75i)T+(39.7+25.0i)T2 1 + (9.52 + 2.75i)T + (39.7 + 25.0i)T^{2}
53 1+(5.16+7.04i)T+(−15.9+50.5i)T2 1 + (5.16 + 7.04i)T + (-15.9 + 50.5i)T^{2}
59 1+(0.0826+0.209i)T+(−43.1+40.2i)T2 1 + (0.0826 + 0.209i)T + (-43.1 + 40.2i)T^{2}
61 1+(3.69+4.56i)T+(−12.7+59.6i)T2 1 + (3.69 + 4.56i)T + (-12.7 + 59.6i)T^{2}
67 1+(4.98+1.96i)T+(49.0+45.6i)T2 1 + (4.98 + 1.96i)T + (49.0 + 45.6i)T^{2}
71 1+(2.74−3.93i)T+(−24.4−66.6i)T2 1 + (2.74 - 3.93i)T + (-24.4 - 66.6i)T^{2}
73 1+(2.57−7.03i)T+(−55.6−47.2i)T2 1 + (2.57 - 7.03i)T + (-55.6 - 47.2i)T^{2}
79 1+(12.8+9.86i)T+(20.1+76.3i)T2 1 + (12.8 + 9.86i)T + (20.1 + 76.3i)T^{2}
83 1+(−0.0249−0.265i)T+(−81.5+15.4i)T2 1 + (-0.0249 - 0.265i)T + (-81.5 + 15.4i)T^{2}
89 1+(−2.54−15.3i)T+(−84.2+28.6i)T2 1 + (-2.54 - 15.3i)T + (-84.2 + 28.6i)T^{2}
97 1+(−4.81+5.94i)T+(−20.3−94.8i)T2 1 + (-4.81 + 5.94i)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

−10.57430278102301203550662515892, −9.844178075047552226089086737445, −8.670428294754028116536418642694, −8.234508053813951231870698265782, −7.985606277205071590804415695245, −5.79252218678074681499777909690, −5.13465098095242025706639940698, −3.18903249976758965491158255067, −3.00263538881983016583255022324, −1.65664510616724219013248683941, 1.55697267392664324392981868514, 3.09767211729463070235768953733, 4.32914289266238749647128909573, 4.97473468462238770467261441417, 6.97906597775417665858623728866, 7.45666886987680683063095131197, 7.917035725661495681054335807769, 9.092050226004085855891539991975, 9.824111993791754033916288459455, 10.48934499786636120060919263854

Graph of the ZZ-function along the critical line