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Properties

Label 2-570-19.16-c1-0-5
Degree 22
Conductor 570570
Sign 0.766+0.642i0.766 + 0.642i
Analytic cond. 4.551474.55147
Root an. cond. 2.133412.13341
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  + (−0.766 − 0.642i)2-s + (−0.939 + 0.342i)3-s + (0.173 + 0.984i)4-s + (0.173 − 0.984i)5-s + (0.939 + 0.342i)6-s + (0.266 + 0.460i)7-s + (0.500 − 0.866i)8-s + (0.766 − 0.642i)9-s + (−0.766 + 0.642i)10-s + (−0.326 + 0.565i)11-s + (−0.499 − 0.866i)12-s + (0.439 + 0.160i)13-s + (0.0923 − 0.524i)14-s + (0.173 + 0.984i)15-s + (−0.939 + 0.342i)16-s + (0.439 + 0.368i)17-s + ⋯
L(s)  = 1  + (−0.541 − 0.454i)2-s + (−0.542 + 0.197i)3-s + (0.0868 + 0.492i)4-s + (0.0776 − 0.440i)5-s + (0.383 + 0.139i)6-s + (0.100 + 0.174i)7-s + (0.176 − 0.306i)8-s + (0.255 − 0.214i)9-s + (−0.242 + 0.203i)10-s + (−0.0983 + 0.170i)11-s + (−0.144 − 0.250i)12-s + (0.121 + 0.0443i)13-s + (0.0246 − 0.140i)14-s + (0.0448 + 0.254i)15-s + (−0.234 + 0.0855i)16-s + (0.106 + 0.0894i)17-s + ⋯

Functional equation

Λ(s)=(570s/2 ΓC(s) L(s)=((0.766+0.642i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(570s/2 ΓC(s+1/2) L(s)=((0.766+0.642i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 570570    =    2⋅3⋅5⋅192 \cdot 3 \cdot 5 \cdot 19
Sign: 0.766+0.642i0.766 + 0.642i
Analytic conductor: 4.551474.55147
Root analytic conductor: 2.133412.13341
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ570(301,⋅)\chi_{570} (301, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 570, ( :1/2), 0.766+0.642i)(2,\ 570,\ (\ :1/2),\ 0.766 + 0.642i)

Particular Values

L(1)L(1) ≈\approx 0.895473−0.325594i0.895473 - 0.325594i
L(12)L(\frac12) ≈\approx 0.895473−0.325594i0.895473 - 0.325594i
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+(0.766+0.642i)T 1 + (0.766 + 0.642i)T
3 1+(0.939−0.342i)T 1 + (0.939 - 0.342i)T
5 1+(−0.173+0.984i)T 1 + (-0.173 + 0.984i)T
19 1+(−4.07−1.55i)T 1 + (-4.07 - 1.55i)T
good7 1+(−0.266−0.460i)T+(−3.5+6.06i)T2 1 + (-0.266 - 0.460i)T + (-3.5 + 6.06i)T^{2}
11 1+(0.326−0.565i)T+(−5.5−9.52i)T2 1 + (0.326 - 0.565i)T + (-5.5 - 9.52i)T^{2}
13 1+(−0.439−0.160i)T+(9.95+8.35i)T2 1 + (-0.439 - 0.160i)T + (9.95 + 8.35i)T^{2}
17 1+(−0.439−0.368i)T+(2.95+16.7i)T2 1 + (-0.439 - 0.368i)T + (2.95 + 16.7i)T^{2}
23 1+(1.09+6.19i)T+(−21.6+7.86i)T2 1 + (1.09 + 6.19i)T + (-21.6 + 7.86i)T^{2}
29 1+(−4.55+3.82i)T+(5.03−28.5i)T2 1 + (-4.55 + 3.82i)T + (5.03 - 28.5i)T^{2}
31 1+(−4.05−7.02i)T+(−15.5+26.8i)T2 1 + (-4.05 - 7.02i)T + (-15.5 + 26.8i)T^{2}
37 1−4.17T+37T2 1 - 4.17T + 37T^{2}
41 1+(−9.99+3.63i)T+(31.4−26.3i)T2 1 + (-9.99 + 3.63i)T + (31.4 - 26.3i)T^{2}
43 1+(−0.578+3.28i)T+(−40.4−14.7i)T2 1 + (-0.578 + 3.28i)T + (-40.4 - 14.7i)T^{2}
47 1+(−3.75+3.15i)T+(8.16−46.2i)T2 1 + (-3.75 + 3.15i)T + (8.16 - 46.2i)T^{2}
53 1+(1.86+10.5i)T+(−49.8+18.1i)T2 1 + (1.86 + 10.5i)T + (-49.8 + 18.1i)T^{2}
59 1+(−7.07−5.93i)T+(10.2+58.1i)T2 1 + (-7.07 - 5.93i)T + (10.2 + 58.1i)T^{2}
61 1+(0.503+2.85i)T+(−57.3+20.8i)T2 1 + (0.503 + 2.85i)T + (-57.3 + 20.8i)T^{2}
67 1+(7.88−6.61i)T+(11.6−65.9i)T2 1 + (7.88 - 6.61i)T + (11.6 - 65.9i)T^{2}
71 1+(1.17−6.67i)T+(−66.7−24.2i)T2 1 + (1.17 - 6.67i)T + (-66.7 - 24.2i)T^{2}
73 1+(0.247−0.0901i)T+(55.9−46.9i)T2 1 + (0.247 - 0.0901i)T + (55.9 - 46.9i)T^{2}
79 1+(−2.49+0.907i)T+(60.5−50.7i)T2 1 + (-2.49 + 0.907i)T + (60.5 - 50.7i)T^{2}
83 1+(1.78+3.09i)T+(−41.5+71.8i)T2 1 + (1.78 + 3.09i)T + (-41.5 + 71.8i)T^{2}
89 1+(3.72+1.35i)T+(68.1+57.2i)T2 1 + (3.72 + 1.35i)T + (68.1 + 57.2i)T^{2}
97 1+(−2.89−2.42i)T+(16.8+95.5i)T2 1 + (-2.89 - 2.42i)T + (16.8 + 95.5i)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

−10.45295301125221404741013592098, −9.982834848975447946778750520534, −8.935168859175879057901491503070, −8.211436989972164418038831277958, −7.12545996505014054088659282945, −6.04709470795218191285457429648, −4.98587695659416134640466701268, −3.96266798603349577457080102312, −2.49273022674091842537828904015, −0.917200015963794142517730741304, 1.10957861463478056120659007418, 2.82587386145345647755846696205, 4.41084535118673877238889793271, 5.61723640639435822383588298476, 6.29461541071027269162351551178, 7.40487314964740248254239244468, 7.86733640996083451275907047053, 9.216793340351153519781004410842, 9.873317691904223564671856762654, 10.88616895869109935654937625539

Graph of the ZZ-function along the critical line