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Properties

Label 2-405-135.122-c1-0-9
Degree 22
Conductor 405405
Sign −0.0200−0.999i-0.0200 - 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  + (2.12 + 1.48i)2-s + (1.60 + 4.41i)4-s + (2.09 + 0.783i)5-s + (−1.25 − 2.68i)7-s + (−1.80 + 6.75i)8-s + (3.27 + 4.77i)10-s + (1.00 + 1.19i)11-s + (−2.29 − 3.28i)13-s + (1.33 − 7.54i)14-s + (−6.66 + 5.59i)16-s + (−0.490 − 1.82i)17-s + (−2.41 + 1.39i)19-s + (−0.0957 + 10.5i)20-s + (0.352 + 4.02i)22-s + (−6.78 − 3.16i)23-s + ⋯
L(s)  = 1  + (1.49 + 1.04i)2-s + (0.803 + 2.20i)4-s + (0.936 + 0.350i)5-s + (−0.473 − 1.01i)7-s + (−0.639 + 2.38i)8-s + (1.03 + 1.50i)10-s + (0.302 + 0.360i)11-s + (−0.637 − 0.910i)13-s + (0.355 − 2.01i)14-s + (−1.66 + 1.39i)16-s + (−0.118 − 0.443i)17-s + (−0.553 + 0.319i)19-s + (−0.0214 + 2.35i)20-s + (0.0751 + 0.859i)22-s + (−1.41 − 0.659i)23-s + ⋯

Functional equation

Λ(s)=(405s/2 ΓC(s) L(s)=((−0.0200−0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0200 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(405s/2 ΓC(s+1/2) L(s)=((−0.0200−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0200 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 405405    =    34⋅53^{4} \cdot 5
Sign: −0.0200−0.999i-0.0200 - 0.999i
Analytic conductor: 3.233943.23394
Root analytic conductor: 1.798311.79831
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ405(152,⋅)\chi_{405} (152, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 405, ( :1/2), −0.0200−0.999i)(2,\ 405,\ (\ :1/2),\ -0.0200 - 0.999i)

Particular Values

L(1)L(1) ≈\approx 2.23650+2.28182i2.23650 + 2.28182i
L(12)L(\frac12) ≈\approx 2.23650+2.28182i2.23650 + 2.28182i
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+(−2.09−0.783i)T 1 + (-2.09 - 0.783i)T
good2 1+(−2.12−1.48i)T+(0.684+1.87i)T2 1 + (-2.12 - 1.48i)T + (0.684 + 1.87i)T^{2}
7 1+(1.25+2.68i)T+(−4.49+5.36i)T2 1 + (1.25 + 2.68i)T + (-4.49 + 5.36i)T^{2}
11 1+(−1.00−1.19i)T+(−1.91+10.8i)T2 1 + (-1.00 - 1.19i)T + (-1.91 + 10.8i)T^{2}
13 1+(2.29+3.28i)T+(−4.44+12.2i)T2 1 + (2.29 + 3.28i)T + (-4.44 + 12.2i)T^{2}
17 1+(0.490+1.82i)T+(−14.7+8.5i)T2 1 + (0.490 + 1.82i)T + (-14.7 + 8.5i)T^{2}
19 1+(2.41−1.39i)T+(9.5−16.4i)T2 1 + (2.41 - 1.39i)T + (9.5 - 16.4i)T^{2}
23 1+(6.78+3.16i)T+(14.7+17.6i)T2 1 + (6.78 + 3.16i)T + (14.7 + 17.6i)T^{2}
29 1+(−0.683−3.87i)T+(−27.2+9.91i)T2 1 + (-0.683 - 3.87i)T + (-27.2 + 9.91i)T^{2}
31 1+(−5.45+1.98i)T+(23.7−19.9i)T2 1 + (-5.45 + 1.98i)T + (23.7 - 19.9i)T^{2}
37 1+(0.316−0.0847i)T+(32.0−18.5i)T2 1 + (0.316 - 0.0847i)T + (32.0 - 18.5i)T^{2}
41 1+(6.26+1.10i)T+(38.5+14.0i)T2 1 + (6.26 + 1.10i)T + (38.5 + 14.0i)T^{2}
43 1+(−0.0509+0.582i)T+(−42.3−7.46i)T2 1 + (-0.0509 + 0.582i)T + (-42.3 - 7.46i)T^{2}
47 1+(−0.690+0.321i)T+(30.2−36.0i)T2 1 + (-0.690 + 0.321i)T + (30.2 - 36.0i)T^{2}
53 1+(−5.57−5.57i)T+53iT2 1 + (-5.57 - 5.57i)T + 53iT^{2}
59 1+(7.84+6.57i)T+(10.2+58.1i)T2 1 + (7.84 + 6.57i)T + (10.2 + 58.1i)T^{2}
61 1+(−11.4−4.15i)T+(46.7+39.2i)T2 1 + (-11.4 - 4.15i)T + (46.7 + 39.2i)T^{2}
67 1+(1.63−1.14i)T+(22.9−62.9i)T2 1 + (1.63 - 1.14i)T + (22.9 - 62.9i)T^{2}
71 1+(−6.11−3.52i)T+(35.5+61.4i)T2 1 + (-6.11 - 3.52i)T + (35.5 + 61.4i)T^{2}
73 1+(−3.93−1.05i)T+(63.2+36.5i)T2 1 + (-3.93 - 1.05i)T + (63.2 + 36.5i)T^{2}
79 1+(12.9−2.28i)T+(74.2−27.0i)T2 1 + (12.9 - 2.28i)T + (74.2 - 27.0i)T^{2}
83 1+(−2.33+3.33i)T+(−28.3−77.9i)T2 1 + (-2.33 + 3.33i)T + (-28.3 - 77.9i)T^{2}
89 1+(4.23+7.33i)T+(−44.5+77.0i)T2 1 + (4.23 + 7.33i)T + (-44.5 + 77.0i)T^{2}
97 1+(−7.95−0.696i)T+(95.5+16.8i)T2 1 + (-7.95 - 0.696i)T + (95.5 + 16.8i)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

−11.93471336273485634342347169384, −10.51544672977249727264290289361, −9.864769787678342687055415814524, −8.317884204248942864441531825122, −7.25775844475911186644639299489, −6.63626950156627305121631142913, −5.80290010612213972571626050680, −4.77140642368508578424100710195, −3.74700951190684204470126782649, −2.55462939363711780091167109132, 1.83059841490269804121193592483, 2.64584728294453937629331205801, 4.02518685965468712386622510208, 5.06250852038482325360294651352, 5.98407788149714966323331583567, 6.53490670896660612738400142153, 8.608388670162324944689589925745, 9.633104585120140294633401635668, 10.18523867238018518877033815467, 11.42652560004171928230722347048

Graph of the ZZ-function along the critical line