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Properties

Label 2-270-135.113-c1-0-1
Degree 22
Conductor 270270
Sign −0.984−0.173i-0.984 - 0.173i
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 + (0.459 + 1.67i)3-s + (0.984 + 0.173i)4-s + (−2.23 − 0.107i)5-s + (−0.312 − 1.70i)6-s + (−1.02 − 0.721i)7-s + (−0.965 − 0.258i)8-s + (−2.57 + 1.53i)9-s + (2.21 + 0.302i)10-s + (−2.11 + 5.81i)11-s + (0.162 + 1.72i)12-s + (−0.354 − 4.04i)13-s + (0.963 + 0.808i)14-s + (−0.846 − 3.77i)15-s + (0.939 + 0.342i)16-s + (−1.07 + 0.287i)17-s + ⋯
L(s)  = 1  + (−0.704 − 0.0616i)2-s + (0.265 + 0.964i)3-s + (0.492 + 0.0868i)4-s + (−0.998 − 0.0482i)5-s + (−0.127 − 0.695i)6-s + (−0.389 − 0.272i)7-s + (−0.341 − 0.0915i)8-s + (−0.859 + 0.511i)9-s + (0.700 + 0.0955i)10-s + (−0.637 + 1.75i)11-s + (0.0469 + 0.497i)12-s + (−0.0982 − 1.12i)13-s + (0.257 + 0.216i)14-s + (−0.218 − 0.975i)15-s + (0.234 + 0.0855i)16-s + (−0.260 + 0.0698i)17-s + ⋯

Functional equation

Λ(s)=(270s/2 ΓC(s) L(s)=((−0.984−0.173i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(270s/2 ΓC(s+1/2) L(s)=((−0.984−0.173i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 270270    =    2⋅33⋅52 \cdot 3^{3} \cdot 5
Sign: −0.984−0.173i-0.984 - 0.173i
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), −0.984−0.173i)(2,\ 270,\ (\ :1/2),\ -0.984 - 0.173i)

Particular Values

L(1)L(1) ≈\approx 0.0309546+0.353833i0.0309546 + 0.353833i
L(12)L(\frac12) ≈\approx 0.0309546+0.353833i0.0309546 + 0.353833i
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+(−0.459−1.67i)T 1 + (-0.459 - 1.67i)T
5 1+(2.23+0.107i)T 1 + (2.23 + 0.107i)T
good7 1+(1.02+0.721i)T+(2.39+6.57i)T2 1 + (1.02 + 0.721i)T + (2.39 + 6.57i)T^{2}
11 1+(2.11−5.81i)T+(−8.42−7.07i)T2 1 + (2.11 - 5.81i)T + (-8.42 - 7.07i)T^{2}
13 1+(0.354+4.04i)T+(−12.8+2.25i)T2 1 + (0.354 + 4.04i)T + (-12.8 + 2.25i)T^{2}
17 1+(1.07−0.287i)T+(14.7−8.5i)T2 1 + (1.07 - 0.287i)T + (14.7 - 8.5i)T^{2}
19 1+(4.49−2.59i)T+(9.5−16.4i)T2 1 + (4.49 - 2.59i)T + (9.5 - 16.4i)T^{2}
23 1+(2.75+3.93i)T+(−7.86+21.6i)T2 1 + (2.75 + 3.93i)T + (-7.86 + 21.6i)T^{2}
29 1+(−0.572+0.480i)T+(5.03−28.5i)T2 1 + (-0.572 + 0.480i)T + (5.03 - 28.5i)T^{2}
31 1+(0.766−4.34i)T+(−29.1−10.6i)T2 1 + (0.766 - 4.34i)T + (-29.1 - 10.6i)T^{2}
37 1+(−1.29−4.82i)T+(−32.0+18.5i)T2 1 + (-1.29 - 4.82i)T + (-32.0 + 18.5i)T^{2}
41 1+(3.72−4.43i)T+(−7.11−40.3i)T2 1 + (3.72 - 4.43i)T + (-7.11 - 40.3i)T^{2}
43 1+(−5.20−11.1i)T+(−27.6+32.9i)T2 1 + (-5.20 - 11.1i)T + (-27.6 + 32.9i)T^{2}
47 1+(−0.919+1.31i)T+(−16.0−44.1i)T2 1 + (-0.919 + 1.31i)T + (-16.0 - 44.1i)T^{2}
53 1+(0.785−0.785i)T−53iT2 1 + (0.785 - 0.785i)T - 53iT^{2}
59 1+(−13.8+5.03i)T+(45.1−37.9i)T2 1 + (-13.8 + 5.03i)T + (45.1 - 37.9i)T^{2}
61 1+(0.115+0.653i)T+(−57.3+20.8i)T2 1 + (0.115 + 0.653i)T + (-57.3 + 20.8i)T^{2}
67 1+(3.62−0.317i)T+(65.9−11.6i)T2 1 + (3.62 - 0.317i)T + (65.9 - 11.6i)T^{2}
71 1+(−9.91−5.72i)T+(35.5+61.4i)T2 1 + (-9.91 - 5.72i)T + (35.5 + 61.4i)T^{2}
73 1+(0.410−1.53i)T+(−63.2−36.5i)T2 1 + (0.410 - 1.53i)T + (-63.2 - 36.5i)T^{2}
79 1+(−4.82−5.74i)T+(−13.7+77.7i)T2 1 + (-4.82 - 5.74i)T + (-13.7 + 77.7i)T^{2}
83 1+(0.846−9.67i)T+(−81.7−14.4i)T2 1 + (0.846 - 9.67i)T + (-81.7 - 14.4i)T^{2}
89 1+(0.959+1.66i)T+(−44.5+77.0i)T2 1 + (0.959 + 1.66i)T + (-44.5 + 77.0i)T^{2}
97 1+(−3.12+1.45i)T+(62.3−74.3i)T2 1 + (-3.12 + 1.45i)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

−12.31087751552513282780668050348, −11.08399329938733849460882360542, −10.25636610636599922207488899951, −9.797209389348181589825224686761, −8.394592707223629892650511882633, −7.897167386541578042145519721343, −6.68411827452824663370697690363, −4.98033549110376322725395695834, −3.97154597842396610796302916503, −2.62454268700022192849792986736, 0.30993204734029730618284459289, 2.40743766971324214125279472850, 3.71793943653912669207109403010, 5.77211042915921914841994785667, 6.75630486833895612073629853086, 7.66629837498445791363078667649, 8.574548609851691439494946869651, 9.101766876575202338601141679933, 10.79494816399835230079649744289, 11.44200166086505728710371059183

Graph of the ZZ-function along the critical line