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Properties

Label 2-490-1.1-c3-0-7
Degree 22
Conductor 490490
Sign 11
Analytic cond. 28.910928.9109
Root an. cond. 5.376885.37688
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3-s + 4·4-s + 5·5-s − 2·6-s − 8·8-s − 26·9-s − 10·10-s − 2·11-s + 4·12-s + 8·13-s + 5·15-s + 16·16-s + 52·17-s + 52·18-s − 26·19-s + 20·20-s + 4·22-s + 67·23-s − 8·24-s + 25·25-s − 16·26-s − 53·27-s + 69·29-s − 10·30-s + 332·31-s − 32·32-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.192·3-s + 1/2·4-s + 0.447·5-s − 0.136·6-s − 0.353·8-s − 0.962·9-s − 0.316·10-s − 0.0548·11-s + 0.0962·12-s + 0.170·13-s + 0.0860·15-s + 1/4·16-s + 0.741·17-s + 0.680·18-s − 0.313·19-s + 0.223·20-s + 0.0387·22-s + 0.607·23-s − 0.0680·24-s + 1/5·25-s − 0.120·26-s − 0.377·27-s + 0.441·29-s − 0.0608·30-s + 1.92·31-s − 0.176·32-s + ⋯

Functional equation

Λ(s)=(490s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(490s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, 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: 28.910928.9109
Root analytic conductor: 5.376885.37688
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 490, ( :3/2), 1)(2,\ 490,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.4901567751.490156775
L(12)L(\frac12) ≈\approx 1.4901567751.490156775
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)
bad2 1+pT 1 + p T
5 1−pT 1 - p T
7 1 1
good3 1−T+p3T2 1 - T + p^{3} T^{2}
11 1+2T+p3T2 1 + 2 T + p^{3} T^{2}
13 1−8T+p3T2 1 - 8 T + p^{3} T^{2}
17 1−52T+p3T2 1 - 52 T + p^{3} T^{2}
19 1+26T+p3T2 1 + 26 T + p^{3} T^{2}
23 1−67T+p3T2 1 - 67 T + p^{3} T^{2}
29 1−69T+p3T2 1 - 69 T + p^{3} T^{2}
31 1−332T+p3T2 1 - 332 T + p^{3} T^{2}
37 1−196T+p3T2 1 - 196 T + p^{3} T^{2}
41 1+353T+p3T2 1 + 353 T + p^{3} T^{2}
43 1+369T+p3T2 1 + 369 T + p^{3} T^{2}
47 1+88T+p3T2 1 + 88 T + p^{3} T^{2}
53 1−582T+p3T2 1 - 582 T + p^{3} T^{2}
59 1−350T+p3T2 1 - 350 T + p^{3} T^{2}
61 1−467T+p3T2 1 - 467 T + p^{3} T^{2}
67 1−291T+p3T2 1 - 291 T + p^{3} T^{2}
71 1−770T+p3T2 1 - 770 T + p^{3} T^{2}
73 1+628T+p3T2 1 + 628 T + p^{3} T^{2}
79 1−1170T+p3T2 1 - 1170 T + p^{3} T^{2}
83 1+525T+p3T2 1 + 525 T + p^{3} T^{2}
89 1+pT+p3T2 1 + p T + p^{3} T^{2}
97 1−290T+p3T2 1 - 290 T + p^{3} 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

−10.31326811407164678881123813400, −9.728138980797757092857073038461, −8.600425776644537930212052514150, −8.195547144602069151204134805798, −6.90660916463151409284539033435, −6.04306578245275794700473369377, −5.00759978637281288340496691006, −3.34471378606230485181948214199, −2.32829999462431807938763059265, −0.841771476100948610717919718941, 0.841771476100948610717919718941, 2.32829999462431807938763059265, 3.34471378606230485181948214199, 5.00759978637281288340496691006, 6.04306578245275794700473369377, 6.90660916463151409284539033435, 8.195547144602069151204134805798, 8.600425776644537930212052514150, 9.728138980797757092857073038461, 10.31326811407164678881123813400

Graph of the ZZ-function along the critical line