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Properties

Label 2-360-9.4-c1-0-11
Degree 22
Conductor 360360
Sign −0.870+0.491i-0.870 + 0.491i
Analytic cond. 2.874612.87461
Root an. cond. 1.695461.69546
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  + (−0.724 − 1.57i)3-s + (0.5 − 0.866i)5-s + (−1.72 − 2.98i)7-s + (−1.94 + 2.28i)9-s + (−1.72 − 0.158i)15-s − 2·17-s − 6.89·19-s + (−3.44 + 4.87i)21-s + (3.72 − 6.45i)23-s + (−0.499 − 0.866i)25-s + (5.00 + 1.41i)27-s + (0.949 + 1.64i)29-s + (−0.550 + 0.953i)31-s − 3.44·35-s − 6·37-s + ⋯
L(s)  = 1  + (−0.418 − 0.908i)3-s + (0.223 − 0.387i)5-s + (−0.651 − 1.12i)7-s + (−0.649 + 0.760i)9-s + (−0.445 − 0.0410i)15-s − 0.485·17-s − 1.58·19-s + (−0.752 + 1.06i)21-s + (0.776 − 1.34i)23-s + (−0.0999 − 0.173i)25-s + (0.962 + 0.272i)27-s + (0.176 + 0.305i)29-s + (−0.0988 + 0.171i)31-s − 0.583·35-s − 0.986·37-s + ⋯

Functional equation

Λ(s)=(360s/2 ΓC(s) L(s)=((−0.870+0.491i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.870 + 0.491i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(360s/2 ΓC(s+1/2) L(s)=((−0.870+0.491i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.870 + 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 360360    =    23⋅32⋅52^{3} \cdot 3^{2} \cdot 5
Sign: −0.870+0.491i-0.870 + 0.491i
Analytic conductor: 2.874612.87461
Root analytic conductor: 1.695461.69546
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ360(121,⋅)\chi_{360} (121, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 360, ( :1/2), −0.870+0.491i)(2,\ 360,\ (\ :1/2),\ -0.870 + 0.491i)

Particular Values

L(1)L(1) ≈\approx 0.205140−0.779973i0.205140 - 0.779973i
L(12)L(\frac12) ≈\approx 0.205140−0.779973i0.205140 - 0.779973i
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 1
3 1+(0.724+1.57i)T 1 + (0.724 + 1.57i)T
5 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
good7 1+(1.72+2.98i)T+(−3.5+6.06i)T2 1 + (1.72 + 2.98i)T + (-3.5 + 6.06i)T^{2}
11 1+(−5.5+9.52i)T2 1 + (-5.5 + 9.52i)T^{2}
13 1+(−6.5−11.2i)T2 1 + (-6.5 - 11.2i)T^{2}
17 1+2T+17T2 1 + 2T + 17T^{2}
19 1+6.89T+19T2 1 + 6.89T + 19T^{2}
23 1+(−3.72+6.45i)T+(−11.5−19.9i)T2 1 + (-3.72 + 6.45i)T + (-11.5 - 19.9i)T^{2}
29 1+(−0.949−1.64i)T+(−14.5+25.1i)T2 1 + (-0.949 - 1.64i)T + (-14.5 + 25.1i)T^{2}
31 1+(0.550−0.953i)T+(−15.5−26.8i)T2 1 + (0.550 - 0.953i)T + (-15.5 - 26.8i)T^{2}
37 1+6T+37T2 1 + 6T + 37T^{2}
41 1+(−4.94+8.57i)T+(−20.5−35.5i)T2 1 + (-4.94 + 8.57i)T + (-20.5 - 35.5i)T^{2}
43 1+(5.89+10.2i)T+(−21.5+37.2i)T2 1 + (5.89 + 10.2i)T + (-21.5 + 37.2i)T^{2}
47 1+(−4.72−8.18i)T+(−23.5+40.7i)T2 1 + (-4.72 - 8.18i)T + (-23.5 + 40.7i)T^{2}
53 1−7.79T+53T2 1 - 7.79T + 53T^{2}
59 1+(−0.550+0.953i)T+(−29.5−51.0i)T2 1 + (-0.550 + 0.953i)T + (-29.5 - 51.0i)T^{2}
61 1+(−1.5−2.59i)T+(−30.5+52.8i)T2 1 + (-1.5 - 2.59i)T + (-30.5 + 52.8i)T^{2}
67 1+(−6.62+11.4i)T+(−33.5−58.0i)T2 1 + (-6.62 + 11.4i)T + (-33.5 - 58.0i)T^{2}
71 1−9.79T+71T2 1 - 9.79T + 71T^{2}
73 1−13.7T+73T2 1 - 13.7T + 73T^{2}
79 1+(−3.44−5.97i)T+(−39.5+68.4i)T2 1 + (-3.44 - 5.97i)T + (-39.5 + 68.4i)T^{2}
83 1+(−2.72−4.71i)T+(−41.5+71.8i)T2 1 + (-2.72 - 4.71i)T + (-41.5 + 71.8i)T^{2}
89 1−2.79T+89T2 1 - 2.79T + 89T^{2}
97 1+(1+1.73i)T+(−48.5+84.0i)T2 1 + (1 + 1.73i)T + (-48.5 + 84.0i)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

−10.77799201073929203241526593190, −10.56543134793175215157768563108, −9.061838482737023859914201070831, −8.213947696406722007585818302209, −6.92719753273848474696197163068, −6.56996510580662521243563312487, −5.21943140144790973909746308195, −3.99071748656861140630327823532, −2.26102702955856286519722017493, −0.55773988127577540184293511907, 2.50688123047966187989500561819, 3.70179882711711695871574156615, 5.03210393698369414987593930865, 5.99258756844737649999135968223, 6.73509164905450659471738177886, 8.410216165895431156306392478615, 9.235493685657644308687218149478, 9.931462300792670418285618445412, 10.92573697691591730852676528775, 11.63112352124049156625588442257

Graph of the ZZ-function along the critical line