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Properties

Label 2-425-5.4-c3-0-69
Degree 22
Conductor 425425
Sign 0.447−0.894i0.447 - 0.894i
Analytic cond. 25.075825.0758
Root an. cond. 5.007575.00757
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.16i·2-s − 9.73i·3-s − 2.01·4-s − 30.8·6-s + 10.5i·7-s − 18.9i·8-s − 67.8·9-s − 15.1·11-s + 19.6i·12-s − 89.5i·13-s + 33.3·14-s − 76.0·16-s − 17i·17-s + 214. i·18-s − 1.82·19-s + ⋯
L(s)  = 1  − 1.11i·2-s − 1.87i·3-s − 0.252·4-s − 2.09·6-s + 0.568i·7-s − 0.836i·8-s − 2.51·9-s − 0.414·11-s + 0.473i·12-s − 1.91i·13-s + 0.636·14-s − 1.18·16-s − 0.242i·17-s + 2.81i·18-s − 0.0220·19-s + ⋯

Functional equation

Λ(s)=(425s/2 ΓC(s) L(s)=((0.447−0.894i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(425s/2 ΓC(s+3/2) L(s)=((0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 425425    =    52⋅175^{2} \cdot 17
Sign: 0.447−0.894i0.447 - 0.894i
Analytic conductor: 25.075825.0758
Root analytic conductor: 5.007575.00757
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ425(324,⋅)\chi_{425} (324, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 425, ( :3/2), 0.447−0.894i)(2,\ 425,\ (\ :3/2),\ 0.447 - 0.894i)

Particular Values

L(2)L(2) ≈\approx 1.1677840151.167784015
L(12)L(\frac12) ≈\approx 1.1677840151.167784015
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)
bad5 1 1
17 1+17iT 1 + 17iT
good2 1+3.16iT−8T2 1 + 3.16iT - 8T^{2}
3 1+9.73iT−27T2 1 + 9.73iT - 27T^{2}
7 1−10.5iT−343T2 1 - 10.5iT - 343T^{2}
11 1+15.1T+1.33e3T2 1 + 15.1T + 1.33e3T^{2}
13 1+89.5iT−2.19e3T2 1 + 89.5iT - 2.19e3T^{2}
19 1+1.82T+6.85e3T2 1 + 1.82T + 6.85e3T^{2}
23 1−170.iT−1.21e4T2 1 - 170. iT - 1.21e4T^{2}
29 1+53.6T+2.43e4T2 1 + 53.6T + 2.43e4T^{2}
31 1−147.T+2.97e4T2 1 - 147.T + 2.97e4T^{2}
37 1−94.8iT−5.06e4T2 1 - 94.8iT - 5.06e4T^{2}
41 1−374.T+6.89e4T2 1 - 374.T + 6.89e4T^{2}
43 1−101.iT−7.95e4T2 1 - 101. iT - 7.95e4T^{2}
47 1+489.iT−1.03e5T2 1 + 489. iT - 1.03e5T^{2}
53 1+84.6iT−1.48e5T2 1 + 84.6iT - 1.48e5T^{2}
59 1+467.T+2.05e5T2 1 + 467.T + 2.05e5T^{2}
61 1+836.T+2.26e5T2 1 + 836.T + 2.26e5T^{2}
67 1−47.4iT−3.00e5T2 1 - 47.4iT - 3.00e5T^{2}
71 1−517.T+3.57e5T2 1 - 517.T + 3.57e5T^{2}
73 1+398.iT−3.89e5T2 1 + 398. iT - 3.89e5T^{2}
79 1+856.T+4.93e5T2 1 + 856.T + 4.93e5T^{2}
83 1+946.iT−5.71e5T2 1 + 946. iT - 5.71e5T^{2}
89 1−765.T+7.04e5T2 1 - 765.T + 7.04e5T^{2}
97 1−124.iT−9.12e5T2 1 - 124. iT - 9.12e5T^{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

−10.30829690996432119780274277680, −9.105278711589299979896163619252, −7.949615696427502313457641176171, −7.42127650436598864269329544650, −6.21254148234651631107421725737, −5.41333780727177049165576659323, −3.20419416598168237899651621749, −2.55114690348990730881620473249, −1.42010519696998312589892517685, −0.37352713833955578900936499307, 2.58919305078461590428585189613, 4.21337788274443501308540401523, 4.60464153821501434997297191028, 5.82350846886286880371075683567, 6.67492405669201261923197665222, 7.905452241322425063279167010077, 8.883199750278427651722006974493, 9.485861887060824170861856839642, 10.67048327701948563483164154861, 11.04784138620443582609946427812

Graph of the ZZ-function along the critical line