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Properties

Label 2-525-1.1-c3-0-16
Degree 22
Conductor 525525
Sign 11
Analytic cond. 30.976030.9760
Root an. cond. 5.565605.56560
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.52·2-s + 3·3-s − 5.66·4-s − 4.58·6-s + 7·7-s + 20.8·8-s + 9·9-s + 51.3·11-s − 16.9·12-s − 87.2·13-s − 10.6·14-s + 13.4·16-s + 80.6·17-s − 13.7·18-s − 29.8·19-s + 21·21-s − 78.4·22-s + 1.71·23-s + 62.6·24-s + 133.·26-s + 27·27-s − 39.6·28-s − 204.·29-s − 150.·31-s − 187.·32-s + 154.·33-s − 123.·34-s + ⋯
L(s)  = 1  − 0.540·2-s + 0.577·3-s − 0.708·4-s − 0.311·6-s + 0.377·7-s + 0.922·8-s + 0.333·9-s + 1.40·11-s − 0.408·12-s − 1.86·13-s − 0.204·14-s + 0.209·16-s + 1.15·17-s − 0.180·18-s − 0.360·19-s + 0.218·21-s − 0.760·22-s + 0.0155·23-s + 0.532·24-s + 1.00·26-s + 0.192·27-s − 0.267·28-s − 1.31·29-s − 0.870·31-s − 1.03·32-s + 0.812·33-s − 0.621·34-s + ⋯

Functional equation

Λ(s)=(525s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(525s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 525525    =    3⋅52⋅73 \cdot 5^{2} \cdot 7
Sign: 11
Analytic conductor: 30.976030.9760
Root analytic conductor: 5.565605.56560
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 525, ( :3/2), 1)(2,\ 525,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.5928948781.592894878
L(12)L(\frac12) ≈\approx 1.5928948781.592894878
L(52)L(\frac{5}{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 1−3T 1 - 3T
5 1 1
7 1−7T 1 - 7T
good2 1+1.52T+8T2 1 + 1.52T + 8T^{2}
11 1−51.3T+1.33e3T2 1 - 51.3T + 1.33e3T^{2}
13 1+87.2T+2.19e3T2 1 + 87.2T + 2.19e3T^{2}
17 1−80.6T+4.91e3T2 1 - 80.6T + 4.91e3T^{2}
19 1+29.8T+6.85e3T2 1 + 29.8T + 6.85e3T^{2}
23 1−1.71T+1.21e4T2 1 - 1.71T + 1.21e4T^{2}
29 1+204.T+2.43e4T2 1 + 204.T + 2.43e4T^{2}
31 1+150.T+2.97e4T2 1 + 150.T + 2.97e4T^{2}
37 1−366.T+5.06e4T2 1 - 366.T + 5.06e4T^{2}
41 1−176.T+6.89e4T2 1 - 176.T + 6.89e4T^{2}
43 1−394.T+7.95e4T2 1 - 394.T + 7.95e4T^{2}
47 1−507.T+1.03e5T2 1 - 507.T + 1.03e5T^{2}
53 1+149.T+1.48e5T2 1 + 149.T + 1.48e5T^{2}
59 1−463.T+2.05e5T2 1 - 463.T + 2.05e5T^{2}
61 1−380.T+2.26e5T2 1 - 380.T + 2.26e5T^{2}
67 1+797.T+3.00e5T2 1 + 797.T + 3.00e5T^{2}
71 1+220.T+3.57e5T2 1 + 220.T + 3.57e5T^{2}
73 1−1.01e3T+3.89e5T2 1 - 1.01e3T + 3.89e5T^{2}
79 1+111.T+4.93e5T2 1 + 111.T + 4.93e5T^{2}
83 1−853.T+5.71e5T2 1 - 853.T + 5.71e5T^{2}
89 1+935.T+7.04e5T2 1 + 935.T + 7.04e5T^{2}
97 1−783.T+9.12e5T2 1 - 783.T + 9.12e5T^{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

−10.07034735379694470920804502425, −9.423837180087403023764266390915, −8.921447812706114594747116051176, −7.68262627349362274219596616691, −7.35095989976259354814582132288, −5.73390103094970890650081308637, −4.58354015178327411564346441714, −3.76438616926355740433122976364, −2.17938860412614093740260228562, −0.851715233936014343985784508710, 0.851715233936014343985784508710, 2.17938860412614093740260228562, 3.76438616926355740433122976364, 4.58354015178327411564346441714, 5.73390103094970890650081308637, 7.35095989976259354814582132288, 7.68262627349362274219596616691, 8.921447812706114594747116051176, 9.423837180087403023764266390915, 10.07034735379694470920804502425

Graph of the ZZ-function along the critical line