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Properties

Label 2-490-1.1-c1-0-9
Degree 22
Conductor 490490
Sign 11
Analytic cond. 3.912663.91266
Root an. cond. 1.978041.97804
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2·3-s + 4-s + 5-s + 2·6-s + 8-s + 9-s + 10-s − 4·11-s + 2·12-s + 2·13-s + 2·15-s + 16-s + 8·17-s + 18-s − 6·19-s + 20-s − 4·22-s − 4·23-s + 2·24-s + 25-s + 2·26-s − 4·27-s − 6·29-s + 2·30-s + 4·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.447·5-s + 0.816·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s + 0.577·12-s + 0.554·13-s + 0.516·15-s + 1/4·16-s + 1.94·17-s + 0.235·18-s − 1.37·19-s + 0.223·20-s − 0.852·22-s − 0.834·23-s + 0.408·24-s + 1/5·25-s + 0.392·26-s − 0.769·27-s − 1.11·29-s + 0.365·30-s + 0.718·31-s + 0.176·32-s + ⋯

Functional equation

Λ(s)=(490s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(490s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 490490    =    2⋅5⋅722 \cdot 5 \cdot 7^{2}
Sign: 11
Analytic conductor: 3.912663.91266
Root analytic conductor: 1.978041.97804
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 490, ( :1/2), 1)(2,\ 490,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.0716494583.071649458
L(12)L(\frac12) ≈\approx 3.0716494583.071649458
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
5 1−T 1 - T
7 1 1
good3 1−2T+pT2 1 - 2 T + p T^{2}
11 1+4T+pT2 1 + 4 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1−8T+pT2 1 - 8 T + p T^{2}
19 1+6T+pT2 1 + 6 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1+10T+pT2 1 + 10 T + p T^{2}
41 1−4T+pT2 1 - 4 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1−4T+pT2 1 - 4 T + p T^{2}
53 1−10T+pT2 1 - 10 T + p T^{2}
59 1+14T+pT2 1 + 14 T + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1−12T+pT2 1 - 12 T + p T^{2}
73 1−4T+pT2 1 - 4 T + p T^{2}
79 1−4T+pT2 1 - 4 T + p T^{2}
83 1+2T+pT2 1 + 2 T + p T^{2}
89 1−8T+pT2 1 - 8 T + p T^{2}
97 1+pT2 1 + 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

−10.78524842246065370835052081258, −10.18076940563670721337529296568, −9.102066376137223109803910203953, −8.118957846682560780166935032553, −7.54815335398458076593938530060, −6.10181652811109734263548661708, −5.34673915974432534760216274251, −3.93281458879405714793180621994, −2.99518962250547346483426263862, −1.97892037150738648722925117261, 1.97892037150738648722925117261, 2.99518962250547346483426263862, 3.93281458879405714793180621994, 5.34673915974432534760216274251, 6.10181652811109734263548661708, 7.54815335398458076593938530060, 8.118957846682560780166935032553, 9.102066376137223109803910203953, 10.18076940563670721337529296568, 10.78524842246065370835052081258

Graph of the ZZ-function along the critical line