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Properties

Label 2-575-1.1-c3-0-57
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  − 3.84·2-s − 4.08·3-s + 6.74·4-s + 15.7·6-s + 27.0·7-s + 4.81·8-s − 10.2·9-s − 14.4·11-s − 27.5·12-s − 9.89·13-s − 104.·14-s − 72.4·16-s − 16.5·17-s + 39.4·18-s − 74.5·19-s − 110.·21-s + 55.4·22-s + 23·23-s − 19.6·24-s + 37.9·26-s + 152.·27-s + 182.·28-s + 202.·29-s − 8.09·31-s + 239.·32-s + 59.0·33-s + 63.5·34-s + ⋯
L(s)  = 1  − 1.35·2-s − 0.786·3-s + 0.843·4-s + 1.06·6-s + 1.46·7-s + 0.212·8-s − 0.380·9-s − 0.395·11-s − 0.663·12-s − 0.211·13-s − 1.98·14-s − 1.13·16-s − 0.236·17-s + 0.517·18-s − 0.900·19-s − 1.15·21-s + 0.537·22-s + 0.208·23-s − 0.167·24-s + 0.286·26-s + 1.08·27-s + 1.23·28-s + 1.29·29-s − 0.0468·31-s + 1.32·32-s + 0.311·33-s + 0.320·34-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+3.84T+8T2 1 + 3.84T + 8T^{2}
3 1+4.08T+27T2 1 + 4.08T + 27T^{2}
7 1−27.0T+343T2 1 - 27.0T + 343T^{2}
11 1+14.4T+1.33e3T2 1 + 14.4T + 1.33e3T^{2}
13 1+9.89T+2.19e3T2 1 + 9.89T + 2.19e3T^{2}
17 1+16.5T+4.91e3T2 1 + 16.5T + 4.91e3T^{2}
19 1+74.5T+6.85e3T2 1 + 74.5T + 6.85e3T^{2}
29 1−202.T+2.43e4T2 1 - 202.T + 2.43e4T^{2}
31 1+8.09T+2.97e4T2 1 + 8.09T + 2.97e4T^{2}
37 1−210.T+5.06e4T2 1 - 210.T + 5.06e4T^{2}
41 1+320.T+6.89e4T2 1 + 320.T + 6.89e4T^{2}
43 1+366.T+7.95e4T2 1 + 366.T + 7.95e4T^{2}
47 1−225.T+1.03e5T2 1 - 225.T + 1.03e5T^{2}
53 1−295.T+1.48e5T2 1 - 295.T + 1.48e5T^{2}
59 1−428.T+2.05e5T2 1 - 428.T + 2.05e5T^{2}
61 1+200.T+2.26e5T2 1 + 200.T + 2.26e5T^{2}
67 1−130.T+3.00e5T2 1 - 130.T + 3.00e5T^{2}
71 1−287.T+3.57e5T2 1 - 287.T + 3.57e5T^{2}
73 1+1.22e3T+3.89e5T2 1 + 1.22e3T + 3.89e5T^{2}
79 1+292.T+4.93e5T2 1 + 292.T + 4.93e5T^{2}
83 1+780.T+5.71e5T2 1 + 780.T + 5.71e5T^{2}
89 1+114.T+7.04e5T2 1 + 114.T + 7.04e5T^{2}
97 1−1.47e3T+9.12e5T2 1 - 1.47e3T + 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

−10.09269814351706533028186108303, −8.677707700834454211708229036292, −8.414215686806282994321219068698, −7.43904142022387227545624874803, −6.44237908365802348909958778674, −5.20248183891237152239623898363, −4.49240669810168794457400657672, −2.41144698408741710232158306295, −1.20984451329184593308914980226, 0, 1.20984451329184593308914980226, 2.41144698408741710232158306295, 4.49240669810168794457400657672, 5.20248183891237152239623898363, 6.44237908365802348909958778674, 7.43904142022387227545624874803, 8.414215686806282994321219068698, 8.677707700834454211708229036292, 10.09269814351706533028186108303

Graph of the ZZ-function along the critical line