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Properties

Label 2-575-1.1-c3-0-76
Degree 22
Conductor 575575
Sign −1-1
Analytic cond. 33.926033.9260
Root an. cond. 5.824615.82461
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  + 0.711·2-s + 0.689·3-s − 7.49·4-s + 0.489·6-s + 19.0·7-s − 11.0·8-s − 26.5·9-s + 50.7·11-s − 5.16·12-s − 45.4·13-s + 13.5·14-s + 52.1·16-s − 53.0·17-s − 18.8·18-s + 58.8·19-s + 13.1·21-s + 36.0·22-s + 23·23-s − 7.59·24-s − 32.3·26-s − 36.8·27-s − 142.·28-s − 70.7·29-s − 286.·31-s + 125.·32-s + 34.9·33-s − 37.7·34-s + ⋯
L(s)  = 1  + 0.251·2-s + 0.132·3-s − 0.936·4-s + 0.0333·6-s + 1.02·7-s − 0.486·8-s − 0.982·9-s + 1.38·11-s − 0.124·12-s − 0.969·13-s + 0.258·14-s + 0.814·16-s − 0.756·17-s − 0.246·18-s + 0.710·19-s + 0.136·21-s + 0.349·22-s + 0.208·23-s − 0.0645·24-s − 0.243·26-s − 0.262·27-s − 0.963·28-s − 0.452·29-s − 1.66·31-s + 0.691·32-s + 0.184·33-s − 0.190·34-s + ⋯

Functional equation

Λ(s)=(575s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(575s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 575575    =    52⋅235^{2} \cdot 23
Sign: −1-1
Analytic conductor: 33.926033.9260
Root analytic conductor: 5.824615.82461
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 575, ( :3/2), −1)(2,\ 575,\ (\ :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)
bad5 1 1
23 1−23T 1 - 23T
good2 1−0.711T+8T2 1 - 0.711T + 8T^{2}
3 1−0.689T+27T2 1 - 0.689T + 27T^{2}
7 1−19.0T+343T2 1 - 19.0T + 343T^{2}
11 1−50.7T+1.33e3T2 1 - 50.7T + 1.33e3T^{2}
13 1+45.4T+2.19e3T2 1 + 45.4T + 2.19e3T^{2}
17 1+53.0T+4.91e3T2 1 + 53.0T + 4.91e3T^{2}
19 1−58.8T+6.85e3T2 1 - 58.8T + 6.85e3T^{2}
29 1+70.7T+2.43e4T2 1 + 70.7T + 2.43e4T^{2}
31 1+286.T+2.97e4T2 1 + 286.T + 2.97e4T^{2}
37 1−175.T+5.06e4T2 1 - 175.T + 5.06e4T^{2}
41 1+95.0T+6.89e4T2 1 + 95.0T + 6.89e4T^{2}
43 1+148.T+7.95e4T2 1 + 148.T + 7.95e4T^{2}
47 1+23.5T+1.03e5T2 1 + 23.5T + 1.03e5T^{2}
53 1+379.T+1.48e5T2 1 + 379.T + 1.48e5T^{2}
59 1+741.T+2.05e5T2 1 + 741.T + 2.05e5T^{2}
61 1−5.17T+2.26e5T2 1 - 5.17T + 2.26e5T^{2}
67 1+974.T+3.00e5T2 1 + 974.T + 3.00e5T^{2}
71 1+336.T+3.57e5T2 1 + 336.T + 3.57e5T^{2}
73 1−317.T+3.89e5T2 1 - 317.T + 3.89e5T^{2}
79 1+577.T+4.93e5T2 1 + 577.T + 4.93e5T^{2}
83 1−225.T+5.71e5T2 1 - 225.T + 5.71e5T^{2}
89 1+1.14e3T+7.04e5T2 1 + 1.14e3T + 7.04e5T^{2}
97 1−147.T+9.12e5T2 1 - 147.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

−9.532172738980418731730914842952, −9.094046840498418782973355597197, −8.249014966735985458394064996305, −7.30214661682020467055729248416, −5.99568440001096561679377144935, −5.07366911280769106181196934714, −4.28909766514303131114496830879, −3.15415010887968857558354501397, −1.60672010936864913147097629751, 0, 1.60672010936864913147097629751, 3.15415010887968857558354501397, 4.28909766514303131114496830879, 5.07366911280769106181196934714, 5.99568440001096561679377144935, 7.30214661682020467055729248416, 8.249014966735985458394064996305, 9.094046840498418782973355597197, 9.532172738980418731730914842952

Graph of the ZZ-function along the critical line