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Properties

Label 2-249690-1.1-c1-0-58
Degree 22
Conductor 249690249690
Sign −1-1
Analytic cond. 1993.781993.78
Root an. cond. 44.651844.6518
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 5-s + 6-s + 7-s + 8-s + 9-s + 10-s − 4·11-s + 12-s − 2·13-s + 14-s + 15-s + 16-s + 2·17-s + 18-s + 4·19-s + 20-s + 21-s − 4·22-s + 8·23-s + 24-s + 25-s − 2·26-s + 27-s + 28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s + 0.288·12-s − 0.554·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.218·21-s − 0.852·22-s + 1.66·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.188·28-s + ⋯

Functional equation

Λ(s)=(249690s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 249690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(249690s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 249690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 249690249690    =    2⋅3⋅5⋅7⋅29⋅412 \cdot 3 \cdot 5 \cdot 7 \cdot 29 \cdot 41
Sign: −1-1
Analytic conductor: 1993.781993.78
Root analytic conductor: 44.651844.6518
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 249690, ( :1/2), −1)(2,\ 249690,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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)
bad2 1−T 1 - T
3 1−T 1 - T
5 1−T 1 - T
7 1−T 1 - T
29 1−T 1 - T
41 1−T 1 - T
good11 1+4T+pT2 1 + 4 T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1−2T+pT2 1 - 2 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1−8T+pT2 1 - 8 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1−6T+pT2 1 - 6 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1+4T+pT2 1 + 4 T + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1+12T+pT2 1 + 12 T + p T^{2}
71 1+8T+pT2 1 + 8 T + p T^{2}
73 1−10T+pT2 1 - 10 T + p T^{2}
79 1+16T+pT2 1 + 16 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1−2T+pT2 1 - 2 T + p T^{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

−13.13639448361112, −12.80474315059502, −12.25477395772213, −11.80426560811348, −11.23496002587705, −10.80824067972697, −10.33211767425462, −9.830642506559774, −9.506691795474031, −8.794285663620364, −8.436668459041383, −7.716811917325564, −7.453671751230429, −7.081527114419158, −6.390720201957589, −5.767379223862129, −5.274248681141707, −5.001558014794440, −4.454897081427510, −3.832012260436157, −3.027693629449370, −2.792733427791996, −2.425142527118976, −1.448035312826937, −1.150179592806201, 0, 1.150179592806201, 1.448035312826937, 2.425142527118976, 2.792733427791996, 3.027693629449370, 3.832012260436157, 4.454897081427510, 5.001558014794440, 5.274248681141707, 5.767379223862129, 6.390720201957589, 7.081527114419158, 7.453671751230429, 7.716811917325564, 8.436668459041383, 8.794285663620364, 9.506691795474031, 9.830642506559774, 10.33211767425462, 10.80824067972697, 11.23496002587705, 11.80426560811348, 12.25477395772213, 12.80474315059502, 13.13639448361112

Graph of the ZZ-function along the critical line