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Properties

Label 2-525-1.1-c3-0-5
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  − 4.70·2-s − 3·3-s + 14.1·4-s + 14.1·6-s − 7·7-s − 28.7·8-s + 9·9-s + 24.5·11-s − 42.3·12-s + 35.0·13-s + 32.9·14-s + 22.1·16-s + 18.4·17-s − 42.3·18-s − 67.4·19-s + 21·21-s − 115.·22-s + 145.·23-s + 86.1·24-s − 164.·26-s − 27·27-s − 98.7·28-s + 214.·29-s − 88.6·31-s + 125.·32-s − 73.7·33-s − 86.5·34-s + ⋯
L(s)  = 1  − 1.66·2-s − 0.577·3-s + 1.76·4-s + 0.959·6-s − 0.377·7-s − 1.26·8-s + 0.333·9-s + 0.674·11-s − 1.01·12-s + 0.747·13-s + 0.628·14-s + 0.345·16-s + 0.262·17-s − 0.554·18-s − 0.813·19-s + 0.218·21-s − 1.12·22-s + 1.32·23-s + 0.732·24-s − 1.24·26-s − 0.192·27-s − 0.666·28-s + 1.37·29-s − 0.513·31-s + 0.694·32-s − 0.389·33-s − 0.436·34-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 0.62848375980.6284837598
L(12)L(\frac12) ≈\approx 0.62848375980.6284837598
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+4.70T+8T2 1 + 4.70T + 8T^{2}
11 1−24.5T+1.33e3T2 1 - 24.5T + 1.33e3T^{2}
13 1−35.0T+2.19e3T2 1 - 35.0T + 2.19e3T^{2}
17 1−18.4T+4.91e3T2 1 - 18.4T + 4.91e3T^{2}
19 1+67.4T+6.85e3T2 1 + 67.4T + 6.85e3T^{2}
23 1−145.T+1.21e4T2 1 - 145.T + 1.21e4T^{2}
29 1−214.T+2.43e4T2 1 - 214.T + 2.43e4T^{2}
31 1+88.6T+2.97e4T2 1 + 88.6T + 2.97e4T^{2}
37 1+162.T+5.06e4T2 1 + 162.T + 5.06e4T^{2}
41 1+337.T+6.89e4T2 1 + 337.T + 6.89e4T^{2}
43 1+122.T+7.95e4T2 1 + 122.T + 7.95e4T^{2}
47 1+354.T+1.03e5T2 1 + 354.T + 1.03e5T^{2}
53 1+676.T+1.48e5T2 1 + 676.T + 1.48e5T^{2}
59 1−501.T+2.05e5T2 1 - 501.T + 2.05e5T^{2}
61 1+708.T+2.26e5T2 1 + 708.T + 2.26e5T^{2}
67 1−907.T+3.00e5T2 1 - 907.T + 3.00e5T^{2}
71 1−430.T+3.57e5T2 1 - 430.T + 3.57e5T^{2}
73 1+41.3T+3.89e5T2 1 + 41.3T + 3.89e5T^{2}
79 1−890.T+4.93e5T2 1 - 890.T + 4.93e5T^{2}
83 1−1.05e3T+5.71e5T2 1 - 1.05e3T + 5.71e5T^{2}
89 1−1.47e3T+7.04e5T2 1 - 1.47e3T + 7.04e5T^{2}
97 1+555.T+9.12e5T2 1 + 555.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

−10.40099230716399718473465553066, −9.514644768178388124076130362418, −8.778035039137146358886626658437, −7.998355039869237451727281040568, −6.72590460896638801413599308830, −6.46481252689732640356209446685, −4.91997094614412997784028654818, −3.36926902953674733975486515684, −1.73711471114374038831705788498, −0.66305326649196143557993196086, 0.66305326649196143557993196086, 1.73711471114374038831705788498, 3.36926902953674733975486515684, 4.91997094614412997784028654818, 6.46481252689732640356209446685, 6.72590460896638801413599308830, 7.998355039869237451727281040568, 8.778035039137146358886626658437, 9.514644768178388124076130362418, 10.40099230716399718473465553066

Graph of the ZZ-function along the critical line