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Properties

Label 2-1620-9.4-c1-0-1
Degree 22
Conductor 16201620
Sign −0.173−0.984i-0.173 - 0.984i
Analytic cond. 12.935712.9357
Root an. cond. 3.596633.59663
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 + (−1 − 1.73i)7-s + (−1 + 1.73i)13-s − 3·17-s + 5·19-s + (−1.5 + 2.59i)23-s + (−0.499 − 0.866i)25-s + (3 + 5.19i)29-s + (−2.5 + 4.33i)31-s + 1.99·35-s + 2·37-s + (−6 + 10.3i)41-s + (−4 − 6.92i)43-s + (6 + 10.3i)47-s + (1.50 − 2.59i)49-s + ⋯
L(s)  = 1  + (−0.223 + 0.387i)5-s + (−0.377 − 0.654i)7-s + (−0.277 + 0.480i)13-s − 0.727·17-s + 1.14·19-s + (−0.312 + 0.541i)23-s + (−0.0999 − 0.173i)25-s + (0.557 + 0.964i)29-s + (−0.449 + 0.777i)31-s + 0.338·35-s + 0.328·37-s + (−0.937 + 1.62i)41-s + (−0.609 − 1.05i)43-s + (0.875 + 1.51i)47-s + (0.214 − 0.371i)49-s + ⋯

Functional equation

Λ(s)=(1620s/2 ΓC(s) L(s)=((−0.173−0.984i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1620s/2 ΓC(s+1/2) L(s)=((−0.173−0.984i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16201620    =    22⋅34⋅52^{2} \cdot 3^{4} \cdot 5
Sign: −0.173−0.984i-0.173 - 0.984i
Analytic conductor: 12.935712.9357
Root analytic conductor: 3.596633.59663
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1620(1081,⋅)\chi_{1620} (1081, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1620, ( :1/2), −0.173−0.984i)(2,\ 1620,\ (\ :1/2),\ -0.173 - 0.984i)

Particular Values

L(1)L(1) ≈\approx 0.99592564810.9959256481
L(12)L(\frac12) ≈\approx 0.99592564810.9959256481
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
5 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
good7 1+(1+1.73i)T+(−3.5+6.06i)T2 1 + (1 + 1.73i)T + (-3.5 + 6.06i)T^{2}
11 1+(−5.5+9.52i)T2 1 + (-5.5 + 9.52i)T^{2}
13 1+(1−1.73i)T+(−6.5−11.2i)T2 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2}
17 1+3T+17T2 1 + 3T + 17T^{2}
19 1−5T+19T2 1 - 5T + 19T^{2}
23 1+(1.5−2.59i)T+(−11.5−19.9i)T2 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2}
29 1+(−3−5.19i)T+(−14.5+25.1i)T2 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{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 1−2T+37T2 1 - 2T + 37T^{2}
41 1+(6−10.3i)T+(−20.5−35.5i)T2 1 + (6 - 10.3i)T + (-20.5 - 35.5i)T^{2}
43 1+(4+6.92i)T+(−21.5+37.2i)T2 1 + (4 + 6.92i)T + (-21.5 + 37.2i)T^{2}
47 1+(−6−10.3i)T+(−23.5+40.7i)T2 1 + (-6 - 10.3i)T + (-23.5 + 40.7i)T^{2}
53 1+3T+53T2 1 + 3T + 53T^{2}
59 1+(3−5.19i)T+(−29.5−51.0i)T2 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2}
61 1+(−3.5−6.06i)T+(−30.5+52.8i)T2 1 + (-3.5 - 6.06i)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 1−12T+71T2 1 - 12T + 71T^{2}
73 1+16T+73T2 1 + 16T + 73T^{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+(−7.5−12.9i)T+(−41.5+71.8i)T2 1 + (-7.5 - 12.9i)T + (-41.5 + 71.8i)T^{2}
89 1+12T+89T2 1 + 12T + 89T^{2}
97 1+(−8−13.8i)T+(−48.5+84.0i)T2 1 + (-8 - 13.8i)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

−9.658046576257505396066782608257, −8.894118974858806846509640377690, −7.922361300288781335385497222834, −7.08543005360257846533748695637, −6.66366405066513570604633639399, −5.51398481501151096020107827863, −4.56164670331893563799858427284, −3.63834738003612573374401987901, −2.77877919166242687809943407586, −1.33428248650444441071186419701, 0.39719415030826636888928434503, 2.06303190498852804834951384172, 3.07280062367362272739357168848, 4.13392715776059607618726701998, 5.11112619081516648838162756916, 5.84411902691293804897692935056, 6.74483944259291106520329136216, 7.67967462989094291475036555323, 8.405355951009289857382128446472, 9.188121844361043554238643821877

Graph of the ZZ-function along the critical line