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Properties

Label 2-520-1.1-c1-0-1
Degree 22
Conductor 520520
Sign 11
Analytic cond. 4.152224.15222
Root an. cond. 2.037692.03769
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.44·3-s + 5-s + 2·7-s + 2.99·9-s − 4.44·11-s − 13-s − 2.44·15-s + 6.89·17-s + 0.449·19-s − 4.89·21-s + 6.44·23-s + 25-s + 4·29-s − 4.44·31-s + 10.8·33-s + 2·35-s + 4.89·37-s + 2.44·39-s + 10.8·41-s + 11.3·43-s + 2.99·45-s − 2·47-s − 3·49-s − 16.8·51-s + 1.10·53-s − 4.44·55-s − 1.10·57-s + ⋯
L(s)  = 1  − 1.41·3-s + 0.447·5-s + 0.755·7-s + 0.999·9-s − 1.34·11-s − 0.277·13-s − 0.632·15-s + 1.67·17-s + 0.103·19-s − 1.06·21-s + 1.34·23-s + 0.200·25-s + 0.742·29-s − 0.799·31-s + 1.89·33-s + 0.338·35-s + 0.805·37-s + 0.392·39-s + 1.70·41-s + 1.73·43-s + 0.447·45-s − 0.291·47-s − 0.428·49-s − 2.36·51-s + 0.151·53-s − 0.599·55-s − 0.145·57-s + ⋯

Functional equation

Λ(s)=(520s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(520s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 520520    =    23⋅5⋅132^{3} \cdot 5 \cdot 13
Sign: 11
Analytic conductor: 4.152224.15222
Root analytic conductor: 2.037692.03769
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 520, ( :1/2), 1)(2,\ 520,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.0025610671.002561067
L(12)L(\frac12) ≈\approx 1.0025610671.002561067
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 1
5 1−T 1 - T
13 1+T 1 + T
good3 1+2.44T+3T2 1 + 2.44T + 3T^{2}
7 1−2T+7T2 1 - 2T + 7T^{2}
11 1+4.44T+11T2 1 + 4.44T + 11T^{2}
17 1−6.89T+17T2 1 - 6.89T + 17T^{2}
19 1−0.449T+19T2 1 - 0.449T + 19T^{2}
23 1−6.44T+23T2 1 - 6.44T + 23T^{2}
29 1−4T+29T2 1 - 4T + 29T^{2}
31 1+4.44T+31T2 1 + 4.44T + 31T^{2}
37 1−4.89T+37T2 1 - 4.89T + 37T^{2}
41 1−10.8T+41T2 1 - 10.8T + 41T^{2}
43 1−11.3T+43T2 1 - 11.3T + 43T^{2}
47 1+2T+47T2 1 + 2T + 47T^{2}
53 1−1.10T+53T2 1 - 1.10T + 53T^{2}
59 1−9.34T+59T2 1 - 9.34T + 59T^{2}
61 1+5.79T+61T2 1 + 5.79T + 61T^{2}
67 1+5.10T+67T2 1 + 5.10T + 67T^{2}
71 1+3.55T+71T2 1 + 3.55T + 71T^{2}
73 1+14.6T+73T2 1 + 14.6T + 73T^{2}
79 1−4.89T+79T2 1 - 4.89T + 79T^{2}
83 1−2T+83T2 1 - 2T + 83T^{2}
89 1−6T+89T2 1 - 6T + 89T^{2}
97 1−7.79T+97T2 1 - 7.79T + 97T^{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

−10.81735829551496376384178067299, −10.35473728271326370463691662290, −9.297786020336508059903442128787, −7.949433548834261159646894082631, −7.24832421898336060270397296698, −5.90694615766181500994260288707, −5.38833856141713142480185624416, −4.62367180616311040522304514495, −2.77598347198424651725791736953, −1.01665628423807691467792718144, 1.01665628423807691467792718144, 2.77598347198424651725791736953, 4.62367180616311040522304514495, 5.38833856141713142480185624416, 5.90694615766181500994260288707, 7.24832421898336060270397296698, 7.949433548834261159646894082631, 9.297786020336508059903442128787, 10.35473728271326370463691662290, 10.81735829551496376384178067299

Graph of the ZZ-function along the critical line