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Properties

Label 2-7200-120.59-c1-0-5
Degree 22
Conductor 72007200
Sign −0.758−0.651i-0.758 - 0.651i
Analytic cond. 57.492257.4922
Root an. cond. 7.582367.58236
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  + 1.41·7-s − 0.191i·11-s + 2.63·13-s − 6.20·17-s − 1.52·19-s − 5.25i·23-s − 0.270·29-s + 6.20i·31-s − 7.61·37-s − 9.22i·41-s + 12.7i·43-s − 3.79i·47-s − 5·49-s + 8.77i·53-s + 10.4i·59-s + ⋯
L(s)  = 1  + 0.534·7-s − 0.0577i·11-s + 0.731·13-s − 1.50·17-s − 0.349·19-s − 1.09i·23-s − 0.0502·29-s + 1.11i·31-s − 1.25·37-s − 1.44i·41-s + 1.94i·43-s − 0.553i·47-s − 0.714·49-s + 1.20i·53-s + 1.36i·59-s + ⋯

Functional equation

Λ(s)=(7200s/2 ΓC(s) L(s)=((−0.758−0.651i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.758 - 0.651i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(7200s/2 ΓC(s+1/2) L(s)=((−0.758−0.651i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.758 - 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 72007200    =    25⋅32⋅522^{5} \cdot 3^{2} \cdot 5^{2}
Sign: −0.758−0.651i-0.758 - 0.651i
Analytic conductor: 57.492257.4922
Root analytic conductor: 7.582367.58236
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ7200(3599,⋅)\chi_{7200} (3599, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 7200, ( :1/2), −0.758−0.651i)(2,\ 7200,\ (\ :1/2),\ -0.758 - 0.651i)

Particular Values

L(1)L(1) ≈\approx 0.63938005270.6393800527
L(12)L(\frac12) ≈\approx 0.63938005270.6393800527
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 1
good7 1−1.41T+7T2 1 - 1.41T + 7T^{2}
11 1+0.191iT−11T2 1 + 0.191iT - 11T^{2}
13 1−2.63T+13T2 1 - 2.63T + 13T^{2}
17 1+6.20T+17T2 1 + 6.20T + 17T^{2}
19 1+1.52T+19T2 1 + 1.52T + 19T^{2}
23 1+5.25iT−23T2 1 + 5.25iT - 23T^{2}
29 1+0.270T+29T2 1 + 0.270T + 29T^{2}
31 1−6.20iT−31T2 1 - 6.20iT - 31T^{2}
37 1+7.61T+37T2 1 + 7.61T + 37T^{2}
41 1+9.22iT−41T2 1 + 9.22iT - 41T^{2}
43 1−12.7iT−43T2 1 - 12.7iT - 43T^{2}
47 1+3.79iT−47T2 1 + 3.79iT - 47T^{2}
53 1−8.77iT−53T2 1 - 8.77iT - 53T^{2}
59 1−10.4iT−59T2 1 - 10.4iT - 59T^{2}
61 1−0.382iT−61T2 1 - 0.382iT - 61T^{2}
67 1−1.72iT−67T2 1 - 1.72iT - 67T^{2}
71 1+9.72T+71T2 1 + 9.72T + 71T^{2}
73 1+5.45iT−73T2 1 + 5.45iT - 73T^{2}
79 1−14.3iT−79T2 1 - 14.3iT - 79T^{2}
83 1−15.2T+83T2 1 - 15.2T + 83T^{2}
89 1−3.56iT−89T2 1 - 3.56iT - 89T^{2}
97 1+7.31iT−97T2 1 + 7.31iT - 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

−8.456388660800203935784362277514, −7.45808833845418641454605148333, −6.72999810329530170746961678443, −6.23372480346535932667058116027, −5.32152881778421451122084005357, −4.57914431349983384602058995357, −4.01637102657994166401686197539, −2.99478961467025484170452512681, −2.13247036356692919975088657282, −1.23197346549825087467927314208, 0.14903675268625692823972079921, 1.56099694597211711779059440032, 2.19396004576758360589734021633, 3.34495622810507375512280206711, 4.04981727365287642220594855597, 4.81181148769869084437126324125, 5.50078639935567749618273260770, 6.36352574175949316793197647128, 6.86112888821149682033728095801, 7.76672805793416283711870559700

Graph of the ZZ-function along the critical line