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Properties

Label 2-490-1.1-c1-0-6
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 + 4-s + 5-s + 8-s − 3·9-s + 10-s + 4·11-s + 6·13-s + 16-s − 2·17-s − 3·18-s + 20-s + 4·22-s + 25-s + 6·26-s + 6·29-s − 8·31-s + 32-s − 2·34-s − 3·36-s − 10·37-s + 40-s − 2·41-s + 4·43-s + 4·44-s − 3·45-s − 8·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s − 9-s + 0.316·10-s + 1.20·11-s + 1.66·13-s + 1/4·16-s − 0.485·17-s − 0.707·18-s + 0.223·20-s + 0.852·22-s + 1/5·25-s + 1.17·26-s + 1.11·29-s − 1.43·31-s + 0.176·32-s − 0.342·34-s − 1/2·36-s − 1.64·37-s + 0.158·40-s − 0.312·41-s + 0.609·43-s + 0.603·44-s − 0.447·45-s − 1.16·47-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 2.3696417842.369641784
L(12)L(\frac12) ≈\approx 2.3696417842.369641784
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+pT2 1 + p T^{2}
11 1−4T+pT2 1 - 4 T + p T^{2}
13 1−6T+pT2 1 - 6 T + p T^{2}
17 1+2T+pT2 1 + 2 T + p T^{2}
19 1+pT2 1 + p T^{2}
23 1+pT2 1 + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1+8T+pT2 1 + 8 T + p T^{2}
37 1+10T+pT2 1 + 10 T + p T^{2}
41 1+2T+pT2 1 + 2 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1+8T+pT2 1 + 8 T + p T^{2}
53 1+2T+pT2 1 + 2 T + p T^{2}
59 1−8T+pT2 1 - 8 T + p T^{2}
61 1−14T+pT2 1 - 14 T + p T^{2}
67 1+12T+pT2 1 + 12 T + p T^{2}
71 1+16T+pT2 1 + 16 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1+8T+pT2 1 + 8 T + p T^{2}
83 1+8T+pT2 1 + 8 T + p T^{2}
89 1+10T+pT2 1 + 10 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

−11.21735898278523397026755152007, −10.27331483782898950038120178987, −8.943481127314352432901190185250, −8.517395214779930601558675802260, −6.94587741637952917868946998696, −6.19981958164339530240516278001, −5.43748531590055688874527891076, −4.08647900838425361258038365025, −3.14888451818473432527395410089, −1.60311130730382042811195419884, 1.60311130730382042811195419884, 3.14888451818473432527395410089, 4.08647900838425361258038365025, 5.43748531590055688874527891076, 6.19981958164339530240516278001, 6.94587741637952917868946998696, 8.517395214779930601558675802260, 8.943481127314352432901190185250, 10.27331483782898950038120178987, 11.21735898278523397026755152007

Graph of the ZZ-function along the critical line