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Properties

Label 2-490-1.1-c3-0-24
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 + 8·3-s + 4·4-s − 5·5-s + 16·6-s + 8·8-s + 37·9-s − 10·10-s + 12·11-s + 32·12-s + 58·13-s − 40·15-s + 16·16-s − 66·17-s + 74·18-s + 100·19-s − 20·20-s + 24·22-s + 132·23-s + 64·24-s + 25·25-s + 116·26-s + 80·27-s − 90·29-s − 80·30-s − 152·31-s + 32·32-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.53·3-s + 1/2·4-s − 0.447·5-s + 1.08·6-s + 0.353·8-s + 1.37·9-s − 0.316·10-s + 0.328·11-s + 0.769·12-s + 1.23·13-s − 0.688·15-s + 1/4·16-s − 0.941·17-s + 0.968·18-s + 1.20·19-s − 0.223·20-s + 0.232·22-s + 1.19·23-s + 0.544·24-s + 1/5·25-s + 0.874·26-s + 0.570·27-s − 0.576·29-s − 0.486·30-s − 0.880·31-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 5.2476580935.247658093
L(12)L(\frac12) ≈\approx 5.2476580935.247658093
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 1−8T+p3T2 1 - 8 T + p^{3} T^{2}
11 1−12T+p3T2 1 - 12 T + p^{3} T^{2}
13 1−58T+p3T2 1 - 58 T + p^{3} T^{2}
17 1+66T+p3T2 1 + 66 T + p^{3} T^{2}
19 1−100T+p3T2 1 - 100 T + p^{3} T^{2}
23 1−132T+p3T2 1 - 132 T + p^{3} T^{2}
29 1+90T+p3T2 1 + 90 T + p^{3} T^{2}
31 1+152T+p3T2 1 + 152 T + p^{3} T^{2}
37 1+34T+p3T2 1 + 34 T + p^{3} T^{2}
41 1−438T+p3T2 1 - 438 T + p^{3} T^{2}
43 1−32T+p3T2 1 - 32 T + p^{3} T^{2}
47 1−204T+p3T2 1 - 204 T + p^{3} T^{2}
53 1−222T+p3T2 1 - 222 T + p^{3} T^{2}
59 1+420T+p3T2 1 + 420 T + p^{3} T^{2}
61 1+902T+p3T2 1 + 902 T + p^{3} T^{2}
67 1+1024T+p3T2 1 + 1024 T + p^{3} T^{2}
71 1−432T+p3T2 1 - 432 T + p^{3} T^{2}
73 1+362T+p3T2 1 + 362 T + p^{3} T^{2}
79 1+160T+p3T2 1 + 160 T + p^{3} T^{2}
83 1+72T+p3T2 1 + 72 T + p^{3} T^{2}
89 1+810T+p3T2 1 + 810 T + p^{3} T^{2}
97 1+1106T+p3T2 1 + 1106 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

−10.73658260281362148619536450940, −9.252623632791610704218499799601, −8.885961350295804892542642103262, −7.74974014986234195049779034566, −7.10609510711787411153545829952, −5.84234121340824604379404952862, −4.42121363232926495205334727802, −3.58659471138166665605015375794, −2.79652536489598075351163586538, −1.41968591770692404719310089490, 1.41968591770692404719310089490, 2.79652536489598075351163586538, 3.58659471138166665605015375794, 4.42121363232926495205334727802, 5.84234121340824604379404952862, 7.10609510711787411153545829952, 7.74974014986234195049779034566, 8.885961350295804892542642103262, 9.252623632791610704218499799601, 10.73658260281362148619536450940

Graph of the ZZ-function along the critical line