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Properties

Label 2-405-135.122-c1-0-11
Degree 22
Conductor 405405
Sign −0.989−0.145i-0.989 - 0.145i
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  + (−1.65 − 1.15i)2-s + (0.704 + 1.93i)4-s + (0.268 − 2.21i)5-s + (0.618 + 1.32i)7-s + (0.0310 − 0.115i)8-s + (−3.00 + 3.35i)10-s + (−1.86 − 2.21i)11-s + (−3.51 − 5.02i)13-s + (0.512 − 2.90i)14-s + (2.97 − 2.49i)16-s + (0.833 + 3.10i)17-s + (3.51 − 2.03i)19-s + (4.48 − 1.04i)20-s + (0.508 + 5.80i)22-s + (−4.58 − 2.14i)23-s + ⋯
L(s)  = 1  + (−1.16 − 0.817i)2-s + (0.352 + 0.967i)4-s + (0.120 − 0.992i)5-s + (0.233 + 0.501i)7-s + (0.0109 − 0.0409i)8-s + (−0.951 + 1.06i)10-s + (−0.560 − 0.668i)11-s + (−0.975 − 1.39i)13-s + (0.136 − 0.776i)14-s + (0.742 − 0.623i)16-s + (0.202 + 0.753i)17-s + (0.806 − 0.465i)19-s + (1.00 − 0.233i)20-s + (0.108 + 1.23i)22-s + (−0.957 − 0.446i)23-s + ⋯

Functional equation

Λ(s)=(405s/2 ΓC(s) L(s)=((−0.989−0.145i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.145i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(405s/2 ΓC(s+1/2) L(s)=((−0.989−0.145i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 - 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 405405    =    34⋅53^{4} \cdot 5
Sign: −0.989−0.145i-0.989 - 0.145i
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), −0.989−0.145i)(2,\ 405,\ (\ :1/2),\ -0.989 - 0.145i)

Particular Values

L(1)L(1) ≈\approx 0.0319389+0.437863i0.0319389 + 0.437863i
L(12)L(\frac12) ≈\approx 0.0319389+0.437863i0.0319389 + 0.437863i
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+(−0.268+2.21i)T 1 + (-0.268 + 2.21i)T
good2 1+(1.65+1.15i)T+(0.684+1.87i)T2 1 + (1.65 + 1.15i)T + (0.684 + 1.87i)T^{2}
7 1+(−0.618−1.32i)T+(−4.49+5.36i)T2 1 + (-0.618 - 1.32i)T + (-4.49 + 5.36i)T^{2}
11 1+(1.86+2.21i)T+(−1.91+10.8i)T2 1 + (1.86 + 2.21i)T + (-1.91 + 10.8i)T^{2}
13 1+(3.51+5.02i)T+(−4.44+12.2i)T2 1 + (3.51 + 5.02i)T + (-4.44 + 12.2i)T^{2}
17 1+(−0.833−3.10i)T+(−14.7+8.5i)T2 1 + (-0.833 - 3.10i)T + (-14.7 + 8.5i)T^{2}
19 1+(−3.51+2.03i)T+(9.5−16.4i)T2 1 + (-3.51 + 2.03i)T + (9.5 - 16.4i)T^{2}
23 1+(4.58+2.14i)T+(14.7+17.6i)T2 1 + (4.58 + 2.14i)T + (14.7 + 17.6i)T^{2}
29 1+(0.368+2.08i)T+(−27.2+9.91i)T2 1 + (0.368 + 2.08i)T + (-27.2 + 9.91i)T^{2}
31 1+(3.20−1.16i)T+(23.7−19.9i)T2 1 + (3.20 - 1.16i)T + (23.7 - 19.9i)T^{2}
37 1+(11.3−3.04i)T+(32.0−18.5i)T2 1 + (11.3 - 3.04i)T + (32.0 - 18.5i)T^{2}
41 1+(0.839+0.147i)T+(38.5+14.0i)T2 1 + (0.839 + 0.147i)T + (38.5 + 14.0i)T^{2}
43 1+(0.0641−0.732i)T+(−42.3−7.46i)T2 1 + (0.0641 - 0.732i)T + (-42.3 - 7.46i)T^{2}
47 1+(2.61−1.21i)T+(30.2−36.0i)T2 1 + (2.61 - 1.21i)T + (30.2 - 36.0i)T^{2}
53 1+(−0.0757−0.0757i)T+53iT2 1 + (-0.0757 - 0.0757i)T + 53iT^{2}
59 1+(−2.89−2.43i)T+(10.2+58.1i)T2 1 + (-2.89 - 2.43i)T + (10.2 + 58.1i)T^{2}
61 1+(4.21+1.53i)T+(46.7+39.2i)T2 1 + (4.21 + 1.53i)T + (46.7 + 39.2i)T^{2}
67 1+(−3.65+2.55i)T+(22.9−62.9i)T2 1 + (-3.65 + 2.55i)T + (22.9 - 62.9i)T^{2}
71 1+(−7.84−4.53i)T+(35.5+61.4i)T2 1 + (-7.84 - 4.53i)T + (35.5 + 61.4i)T^{2}
73 1+(8.04+2.15i)T+(63.2+36.5i)T2 1 + (8.04 + 2.15i)T + (63.2 + 36.5i)T^{2}
79 1+(−1.44+0.254i)T+(74.2−27.0i)T2 1 + (-1.44 + 0.254i)T + (74.2 - 27.0i)T^{2}
83 1+(−9.69+13.8i)T+(−28.3−77.9i)T2 1 + (-9.69 + 13.8i)T + (-28.3 - 77.9i)T^{2}
89 1+(3.05+5.28i)T+(−44.5+77.0i)T2 1 + (3.05 + 5.28i)T + (-44.5 + 77.0i)T^{2}
97 1+(−16.1−1.41i)T+(95.5+16.8i)T2 1 + (-16.1 - 1.41i)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

−10.47176628619153884260689529538, −9.995830771145236325616453252293, −8.948906844722794208593664511131, −8.300294303225093726524358843021, −7.65163042617805783906505503139, −5.71466252316868071607663540218, −5.07309929381557275018339586035, −3.20043971255025679460830107644, −1.94361646514324600142641366009, −0.40224370568395652211551868479, 1.99177792850579375813037325540, 3.71558397919037740708706520934, 5.23749960536582014368138098676, 6.56958536611748727985331428903, 7.35681002105533185966491854961, 7.64506772158002721986605758834, 9.086688074423638535685030923041, 9.842575274211322582018552017364, 10.37420996491271573436730177789, 11.51281849932363451930863773941

Graph of the ZZ-function along the critical line