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Properties

Label 2-405-135.113-c1-0-14
Degree 22
Conductor 405405
Sign −0.0316+0.999i-0.0316 + 0.999i
Analytic cond. 3.233943.23394
Root an. cond. 1.798311.79831
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  + (1.36 + 0.119i)2-s + (−0.116 − 0.0204i)4-s + (0.119 − 2.23i)5-s + (−2.68 − 1.87i)7-s + (−2.80 − 0.752i)8-s + (0.430 − 3.03i)10-s + (1.85 − 5.10i)11-s + (0.215 + 2.45i)13-s + (−3.44 − 2.88i)14-s + (−3.52 − 1.28i)16-s + (3.45 − 0.925i)17-s + (0.417 − 0.240i)19-s + (−0.0595 + 0.256i)20-s + (3.14 − 6.75i)22-s + (4.01 + 5.73i)23-s + ⋯
L(s)  = 1  + (0.966 + 0.0845i)2-s + (−0.0580 − 0.0102i)4-s + (0.0533 − 0.998i)5-s + (−1.01 − 0.710i)7-s + (−0.992 − 0.265i)8-s + (0.136 − 0.960i)10-s + (0.560 − 1.53i)11-s + (0.0596 + 0.682i)13-s + (−0.920 − 0.772i)14-s + (−0.881 − 0.320i)16-s + (0.837 − 0.224i)17-s + (0.0957 − 0.0552i)19-s + (−0.0133 + 0.0574i)20-s + (0.671 − 1.43i)22-s + (0.836 + 1.19i)23-s + ⋯

Functional equation

Λ(s)=(405s/2 ΓC(s) L(s)=((−0.0316+0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0316 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(405s/2 ΓC(s+1/2) L(s)=((−0.0316+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0316 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 405405    =    34⋅53^{4} \cdot 5
Sign: −0.0316+0.999i-0.0316 + 0.999i
Analytic conductor: 3.233943.23394
Root analytic conductor: 1.798311.79831
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ405(368,⋅)\chi_{405} (368, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 405, ( :1/2), −0.0316+0.999i)(2,\ 405,\ (\ :1/2),\ -0.0316 + 0.999i)

Particular Values

L(1)L(1) ≈\approx 1.12393−1.16010i1.12393 - 1.16010i
L(12)L(\frac12) ≈\approx 1.12393−1.16010i1.12393 - 1.16010i
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)
bad3 1 1
5 1+(−0.119+2.23i)T 1 + (-0.119 + 2.23i)T
good2 1+(−1.36−0.119i)T+(1.96+0.347i)T2 1 + (-1.36 - 0.119i)T + (1.96 + 0.347i)T^{2}
7 1+(2.68+1.87i)T+(2.39+6.57i)T2 1 + (2.68 + 1.87i)T + (2.39 + 6.57i)T^{2}
11 1+(−1.85+5.10i)T+(−8.42−7.07i)T2 1 + (-1.85 + 5.10i)T + (-8.42 - 7.07i)T^{2}
13 1+(−0.215−2.45i)T+(−12.8+2.25i)T2 1 + (-0.215 - 2.45i)T + (-12.8 + 2.25i)T^{2}
17 1+(−3.45+0.925i)T+(14.7−8.5i)T2 1 + (-3.45 + 0.925i)T + (14.7 - 8.5i)T^{2}
19 1+(−0.417+0.240i)T+(9.5−16.4i)T2 1 + (-0.417 + 0.240i)T + (9.5 - 16.4i)T^{2}
23 1+(−4.01−5.73i)T+(−7.86+21.6i)T2 1 + (-4.01 - 5.73i)T + (-7.86 + 21.6i)T^{2}
29 1+(0.0993−0.0833i)T+(5.03−28.5i)T2 1 + (0.0993 - 0.0833i)T + (5.03 - 28.5i)T^{2}
31 1+(−0.509+2.88i)T+(−29.1−10.6i)T2 1 + (-0.509 + 2.88i)T + (-29.1 - 10.6i)T^{2}
37 1+(−1.67−6.26i)T+(−32.0+18.5i)T2 1 + (-1.67 - 6.26i)T + (-32.0 + 18.5i)T^{2}
41 1+(0.215−0.256i)T+(−7.11−40.3i)T2 1 + (0.215 - 0.256i)T + (-7.11 - 40.3i)T^{2}
43 1+(2.45+5.27i)T+(−27.6+32.9i)T2 1 + (2.45 + 5.27i)T + (-27.6 + 32.9i)T^{2}
47 1+(−6.80+9.71i)T+(−16.0−44.1i)T2 1 + (-6.80 + 9.71i)T + (-16.0 - 44.1i)T^{2}
53 1+(−0.167+0.167i)T−53iT2 1 + (-0.167 + 0.167i)T - 53iT^{2}
59 1+(−3.62+1.32i)T+(45.1−37.9i)T2 1 + (-3.62 + 1.32i)T + (45.1 - 37.9i)T^{2}
61 1+(0.855+4.85i)T+(−57.3+20.8i)T2 1 + (0.855 + 4.85i)T + (-57.3 + 20.8i)T^{2}
67 1+(−4.25+0.372i)T+(65.9−11.6i)T2 1 + (-4.25 + 0.372i)T + (65.9 - 11.6i)T^{2}
71 1+(7.10+4.10i)T+(35.5+61.4i)T2 1 + (7.10 + 4.10i)T + (35.5 + 61.4i)T^{2}
73 1+(3.51−13.1i)T+(−63.2−36.5i)T2 1 + (3.51 - 13.1i)T + (-63.2 - 36.5i)T^{2}
79 1+(−6.44−7.68i)T+(−13.7+77.7i)T2 1 + (-6.44 - 7.68i)T + (-13.7 + 77.7i)T^{2}
83 1+(0.00128−0.0147i)T+(−81.7−14.4i)T2 1 + (0.00128 - 0.0147i)T + (-81.7 - 14.4i)T^{2}
89 1+(−2.55−4.43i)T+(−44.5+77.0i)T2 1 + (-2.55 - 4.43i)T + (-44.5 + 77.0i)T^{2}
97 1+(−3.77+1.75i)T+(62.3−74.3i)T2 1 + (-3.77 + 1.75i)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

−11.39565941741432138277106657643, −9.910320742238438589025859272371, −9.238968747704411594855651977558, −8.428928401775509291085978175362, −6.97302105237956893252642777138, −5.97989903907316855530613732670, −5.21019896235707241373999936652, −3.94534828288221950468432048663, −3.33150518012964986064903342424, −0.808356813514134886802002065448, 2.55913681700033657167755101910, 3.37354403552302781057840167144, 4.54698607096700825346911697735, 5.77640924719070583254153610036, 6.49717143264732401954204284543, 7.51199804698055596730578531391, 8.959297187703191317367847971762, 9.732638010266989990693150939137, 10.57169338685208125893003709310, 11.84629581198131580448529680350

Graph of the ZZ-function along the critical line