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Properties

Label 2-525-1.1-c3-0-28
Degree 22
Conductor 525525
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.53·2-s − 3·3-s + 12.5·4-s + 13.5·6-s + 7·7-s − 20.5·8-s + 9·9-s − 19.0·11-s − 37.5·12-s + 2.93·13-s − 31.7·14-s − 7.21·16-s + 6.49·17-s − 40.7·18-s − 5.43·19-s − 21·21-s + 86.3·22-s − 49.3·23-s + 61.5·24-s − 13.3·26-s − 27·27-s + 87.7·28-s − 291.·29-s + 244.·31-s + 196.·32-s + 57.1·33-s − 29.4·34-s + ⋯
L(s)  = 1  − 1.60·2-s − 0.577·3-s + 1.56·4-s + 0.924·6-s + 0.377·7-s − 0.907·8-s + 0.333·9-s − 0.522·11-s − 0.904·12-s + 0.0626·13-s − 0.605·14-s − 0.112·16-s + 0.0927·17-s − 0.533·18-s − 0.0656·19-s − 0.218·21-s + 0.837·22-s − 0.447·23-s + 0.523·24-s − 0.100·26-s − 0.192·27-s + 0.592·28-s − 1.86·29-s + 1.41·31-s + 1.08·32-s + 0.301·33-s − 0.148·34-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: −1-1
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: 11
Selberg data: (2, 525, ( :3/2), −1)(2,\ 525,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
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+4.53T+8T2 1 + 4.53T + 8T^{2}
11 1+19.0T+1.33e3T2 1 + 19.0T + 1.33e3T^{2}
13 1−2.93T+2.19e3T2 1 - 2.93T + 2.19e3T^{2}
17 1−6.49T+4.91e3T2 1 - 6.49T + 4.91e3T^{2}
19 1+5.43T+6.85e3T2 1 + 5.43T + 6.85e3T^{2}
23 1+49.3T+1.21e4T2 1 + 49.3T + 1.21e4T^{2}
29 1+291.T+2.43e4T2 1 + 291.T + 2.43e4T^{2}
31 1−244.T+2.97e4T2 1 - 244.T + 2.97e4T^{2}
37 1−193.T+5.06e4T2 1 - 193.T + 5.06e4T^{2}
41 1−315.T+6.89e4T2 1 - 315.T + 6.89e4T^{2}
43 1−300.T+7.95e4T2 1 - 300.T + 7.95e4T^{2}
47 1+86.5T+1.03e5T2 1 + 86.5T + 1.03e5T^{2}
53 1+509.T+1.48e5T2 1 + 509.T + 1.48e5T^{2}
59 1+83.3T+2.05e5T2 1 + 83.3T + 2.05e5T^{2}
61 1+5.25T+2.26e5T2 1 + 5.25T + 2.26e5T^{2}
67 1+205.T+3.00e5T2 1 + 205.T + 3.00e5T^{2}
71 1−1.00e3T+3.57e5T2 1 - 1.00e3T + 3.57e5T^{2}
73 1−1.00e3T+3.89e5T2 1 - 1.00e3T + 3.89e5T^{2}
79 1+863.T+4.93e5T2 1 + 863.T + 4.93e5T^{2}
83 1+1.33e3T+5.71e5T2 1 + 1.33e3T + 5.71e5T^{2}
89 1−326.T+7.04e5T2 1 - 326.T + 7.04e5T^{2}
97 1+1.52e3T+9.12e5T2 1 + 1.52e3T + 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

−9.866617179104076557473098780024, −9.292910589293167707581866849496, −8.117656687226005390987407264881, −7.65155649499516237211362880703, −6.58582852725826489533600907087, −5.56991007286448801804177232082, −4.28137017897688057816930304822, −2.46535011381250003962058324278, −1.22479267346348945102799469931, 0, 1.22479267346348945102799469931, 2.46535011381250003962058324278, 4.28137017897688057816930304822, 5.56991007286448801804177232082, 6.58582852725826489533600907087, 7.65155649499516237211362880703, 8.117656687226005390987407264881, 9.292910589293167707581866849496, 9.866617179104076557473098780024

Graph of the ZZ-function along the critical line