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Properties

Label 2-504-7.4-c1-0-1
Degree 22
Conductor 504504
Sign −0.266−0.963i-0.266 - 0.963i
Analytic cond. 4.024464.02446
Root an. cond. 2.006102.00610
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  + (0.5 + 0.866i)5-s + (−2.5 + 0.866i)7-s + (−2.5 + 4.33i)11-s + 2·13-s + (−3 + 5.19i)17-s + (−1 − 1.73i)19-s + (3 + 5.19i)23-s + (2 − 3.46i)25-s − 3·29-s + (−2.5 + 4.33i)31-s + (−2 − 1.73i)35-s + (1 + 1.73i)37-s + 8·41-s − 4·43-s + (−2 − 3.46i)47-s + ⋯
L(s)  = 1  + (0.223 + 0.387i)5-s + (−0.944 + 0.327i)7-s + (−0.753 + 1.30i)11-s + 0.554·13-s + (−0.727 + 1.26i)17-s + (−0.229 − 0.397i)19-s + (0.625 + 1.08i)23-s + (0.400 − 0.692i)25-s − 0.557·29-s + (−0.449 + 0.777i)31-s + (−0.338 − 0.292i)35-s + (0.164 + 0.284i)37-s + 1.24·41-s − 0.609·43-s + (−0.291 − 0.505i)47-s + ⋯

Functional equation

Λ(s)=(504s/2 ΓC(s) L(s)=((−0.266−0.963i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(504s/2 ΓC(s+1/2) L(s)=((−0.266−0.963i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 504504    =    23⋅32⋅72^{3} \cdot 3^{2} \cdot 7
Sign: −0.266−0.963i-0.266 - 0.963i
Analytic conductor: 4.024464.02446
Root analytic conductor: 2.006102.00610
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ504(361,⋅)\chi_{504} (361, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 504, ( :1/2), −0.266−0.963i)(2,\ 504,\ (\ :1/2),\ -0.266 - 0.963i)

Particular Values

L(1)L(1) ≈\approx 0.607340+0.798338i0.607340 + 0.798338i
L(12)L(\frac12) ≈\approx 0.607340+0.798338i0.607340 + 0.798338i
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 1
7 1+(2.5−0.866i)T 1 + (2.5 - 0.866i)T
good5 1+(−0.5−0.866i)T+(−2.5+4.33i)T2 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2}
11 1+(2.5−4.33i)T+(−5.5−9.52i)T2 1 + (2.5 - 4.33i)T + (-5.5 - 9.52i)T^{2}
13 1−2T+13T2 1 - 2T + 13T^{2}
17 1+(3−5.19i)T+(−8.5−14.7i)T2 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2}
19 1+(1+1.73i)T+(−9.5+16.4i)T2 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2}
23 1+(−3−5.19i)T+(−11.5+19.9i)T2 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2}
29 1+3T+29T2 1 + 3T + 29T^{2}
31 1+(2.5−4.33i)T+(−15.5−26.8i)T2 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2}
37 1+(−1−1.73i)T+(−18.5+32.0i)T2 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2}
41 1−8T+41T2 1 - 8T + 41T^{2}
43 1+4T+43T2 1 + 4T + 43T^{2}
47 1+(2+3.46i)T+(−23.5+40.7i)T2 1 + (2 + 3.46i)T + (-23.5 + 40.7i)T^{2}
53 1+(4.5−7.79i)T+(−26.5−45.8i)T2 1 + (4.5 - 7.79i)T + (-26.5 - 45.8i)T^{2}
59 1+(−1.5+2.59i)T+(−29.5−51.0i)T2 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2}
61 1+(−6−10.3i)T+(−30.5+52.8i)T2 1 + (-6 - 10.3i)T + (-30.5 + 52.8i)T^{2}
67 1+(1−1.73i)T+(−33.5−58.0i)T2 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2}
71 1+8T+71T2 1 + 8T + 71T^{2}
73 1+(−7+12.1i)T+(−36.5−63.2i)T2 1 + (-7 + 12.1i)T + (-36.5 - 63.2i)T^{2}
79 1+(0.5+0.866i)T+(−39.5+68.4i)T2 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2}
83 1−17T+83T2 1 - 17T + 83T^{2}
89 1+(9+15.5i)T+(−44.5+77.0i)T2 1 + (9 + 15.5i)T + (-44.5 + 77.0i)T^{2}
97 1−3T+97T2 1 - 3T + 97T^{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

−10.95634341454770104950836846532, −10.33153665707918213721554771598, −9.463200845042747180078195950763, −8.627389753104547939759017785006, −7.40941563614987038053699705727, −6.60662639000712338966927795390, −5.71777337773346049355343596544, −4.46963355997569490413701889847, −3.20927906090790748271329320923, −2.02216694401315799934367373563, 0.57935378938311712888551342100, 2.64449483988025639704890954775, 3.70610374434259922435854071636, 5.05621067840479532079886702507, 6.02071897137359704024113521651, 6.90028831221768740871836491647, 8.047871588492018681910055240308, 8.983874708821005765724252391969, 9.639249753096445109050294396276, 10.83938473008914582192044393326

Graph of the ZZ-function along the critical line