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Properties

Label 2-270-135.113-c1-0-17
Degree 22
Conductor 270270
Sign −0.680+0.732i-0.680 + 0.732i
Analytic cond. 2.155962.15596
Root an. cond. 1.468311.46831
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  + (0.996 + 0.0871i)2-s + (−1.08 − 1.35i)3-s + (0.984 + 0.173i)4-s + (−2.15 − 0.611i)5-s + (−0.963 − 1.43i)6-s + (−3.60 − 2.52i)7-s + (0.965 + 0.258i)8-s + (−0.646 + 2.92i)9-s + (−2.08 − 0.796i)10-s + (0.474 − 1.30i)11-s + (−0.833 − 1.51i)12-s + (−0.410 − 4.69i)13-s + (−3.36 − 2.82i)14-s + (1.50 + 3.56i)15-s + (0.939 + 0.342i)16-s + (5.62 − 1.50i)17-s + ⋯
L(s)  = 1  + (0.704 + 0.0616i)2-s + (−0.626 − 0.779i)3-s + (0.492 + 0.0868i)4-s + (−0.961 − 0.273i)5-s + (−0.393 − 0.587i)6-s + (−1.36 − 0.953i)7-s + (0.341 + 0.0915i)8-s + (−0.215 + 0.976i)9-s + (−0.660 − 0.251i)10-s + (0.143 − 0.393i)11-s + (−0.240 − 0.438i)12-s + (−0.113 − 1.30i)13-s + (−0.900 − 0.755i)14-s + (0.389 + 0.921i)15-s + (0.234 + 0.0855i)16-s + (1.36 − 0.365i)17-s + ⋯

Functional equation

Λ(s)=(270s/2 ΓC(s) L(s)=((−0.680+0.732i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.680 + 0.732i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(270s/2 ΓC(s+1/2) L(s)=((−0.680+0.732i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.680 + 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 270270    =    2⋅33⋅52 \cdot 3^{3} \cdot 5
Sign: −0.680+0.732i-0.680 + 0.732i
Analytic conductor: 2.155962.15596
Root analytic conductor: 1.468311.46831
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ270(113,⋅)\chi_{270} (113, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 270, ( :1/2), −0.680+0.732i)(2,\ 270,\ (\ :1/2),\ -0.680 + 0.732i)

Particular Values

L(1)L(1) ≈\approx 0.363306−0.833684i0.363306 - 0.833684i
L(12)L(\frac12) ≈\approx 0.363306−0.833684i0.363306 - 0.833684i
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+(−0.996−0.0871i)T 1 + (-0.996 - 0.0871i)T
3 1+(1.08+1.35i)T 1 + (1.08 + 1.35i)T
5 1+(2.15+0.611i)T 1 + (2.15 + 0.611i)T
good7 1+(3.60+2.52i)T+(2.39+6.57i)T2 1 + (3.60 + 2.52i)T + (2.39 + 6.57i)T^{2}
11 1+(−0.474+1.30i)T+(−8.42−7.07i)T2 1 + (-0.474 + 1.30i)T + (-8.42 - 7.07i)T^{2}
13 1+(0.410+4.69i)T+(−12.8+2.25i)T2 1 + (0.410 + 4.69i)T + (-12.8 + 2.25i)T^{2}
17 1+(−5.62+1.50i)T+(14.7−8.5i)T2 1 + (-5.62 + 1.50i)T + (14.7 - 8.5i)T^{2}
19 1+(7.17−4.14i)T+(9.5−16.4i)T2 1 + (7.17 - 4.14i)T + (9.5 - 16.4i)T^{2}
23 1+(−0.511−0.730i)T+(−7.86+21.6i)T2 1 + (-0.511 - 0.730i)T + (-7.86 + 21.6i)T^{2}
29 1+(−4.04+3.39i)T+(5.03−28.5i)T2 1 + (-4.04 + 3.39i)T + (5.03 - 28.5i)T^{2}
31 1+(−1.07+6.11i)T+(−29.1−10.6i)T2 1 + (-1.07 + 6.11i)T + (-29.1 - 10.6i)T^{2}
37 1+(−0.319−1.19i)T+(−32.0+18.5i)T2 1 + (-0.319 - 1.19i)T + (-32.0 + 18.5i)T^{2}
41 1+(−1.37+1.63i)T+(−7.11−40.3i)T2 1 + (-1.37 + 1.63i)T + (-7.11 - 40.3i)T^{2}
43 1+(1.59+3.41i)T+(−27.6+32.9i)T2 1 + (1.59 + 3.41i)T + (-27.6 + 32.9i)T^{2}
47 1+(−2.26+3.23i)T+(−16.0−44.1i)T2 1 + (-2.26 + 3.23i)T + (-16.0 - 44.1i)T^{2}
53 1+(6.56−6.56i)T−53iT2 1 + (6.56 - 6.56i)T - 53iT^{2}
59 1+(−2.97+1.08i)T+(45.1−37.9i)T2 1 + (-2.97 + 1.08i)T + (45.1 - 37.9i)T^{2}
61 1+(0.104+0.590i)T+(−57.3+20.8i)T2 1 + (0.104 + 0.590i)T + (-57.3 + 20.8i)T^{2}
67 1+(1.40−0.122i)T+(65.9−11.6i)T2 1 + (1.40 - 0.122i)T + (65.9 - 11.6i)T^{2}
71 1+(−10.7−6.20i)T+(35.5+61.4i)T2 1 + (-10.7 - 6.20i)T + (35.5 + 61.4i)T^{2}
73 1+(−0.128+0.481i)T+(−63.2−36.5i)T2 1 + (-0.128 + 0.481i)T + (-63.2 - 36.5i)T^{2}
79 1+(7.34+8.75i)T+(−13.7+77.7i)T2 1 + (7.34 + 8.75i)T + (-13.7 + 77.7i)T^{2}
83 1+(−0.841+9.61i)T+(−81.7−14.4i)T2 1 + (-0.841 + 9.61i)T + (-81.7 - 14.4i)T^{2}
89 1+(−3.02−5.24i)T+(−44.5+77.0i)T2 1 + (-3.02 - 5.24i)T + (-44.5 + 77.0i)T^{2}
97 1+(3.52−1.64i)T+(62.3−74.3i)T2 1 + (3.52 - 1.64i)T + (62.3 - 74.3i)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

−11.92312675806580038826100473802, −10.76666078120617935762184161709, −10.08202114821796053046259995496, −8.176782999389268547481426287593, −7.51933591942741353971890489406, −6.48731179652568632426788034547, −5.63145655185429090254083192167, −4.15843880630540749803628540220, −3.10085911677825735997097379896, −0.59429507533099108736138336568, 2.91620345588969918315596789349, 3.95592058770276953320880262225, 4.95498370890109934075471069652, 6.35309978308886628961519990178, 6.80283905413211652182971751414, 8.613139990146547106608500002878, 9.573433838259141062565574080070, 10.56506632435701292739516634633, 11.49173578541459841350980845987, 12.36719710458014738829163640996

Graph of the ZZ-function along the critical line