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Properties

Label 2-2175-1.1-c3-0-50
Degree 22
Conductor 21752175
Sign 11
Analytic cond. 128.329128.329
Root an. cond. 11.328211.3282
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.73·2-s − 3·3-s + 5.94·4-s − 11.2·6-s − 11.3·7-s − 7.67·8-s + 9·9-s − 55.6·11-s − 17.8·12-s + 20.9·13-s − 42.5·14-s − 76.2·16-s + 29.8·17-s + 33.6·18-s + 25.7·19-s + 34.1·21-s − 207.·22-s − 32.9·23-s + 23.0·24-s + 78.0·26-s − 27·27-s − 67.7·28-s − 29·29-s − 281.·31-s − 223.·32-s + 166.·33-s + 111.·34-s + ⋯
L(s)  = 1  + 1.32·2-s − 0.577·3-s + 0.743·4-s − 0.762·6-s − 0.615·7-s − 0.339·8-s + 0.333·9-s − 1.52·11-s − 0.429·12-s + 0.446·13-s − 0.812·14-s − 1.19·16-s + 0.425·17-s + 0.440·18-s + 0.311·19-s + 0.355·21-s − 2.01·22-s − 0.298·23-s + 0.195·24-s + 0.588·26-s − 0.192·27-s − 0.457·28-s − 0.185·29-s − 1.63·31-s − 1.23·32-s + 0.880·33-s + 0.561·34-s + ⋯

Functional equation

Λ(s)=(2175s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(2175s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21752175    =    3⋅52⋅293 \cdot 5^{2} \cdot 29
Sign: 11
Analytic conductor: 128.329128.329
Root analytic conductor: 11.328211.3282
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2175, ( :3/2), 1)(2,\ 2175,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.0146605082.014660508
L(12)L(\frac12) ≈\approx 2.0146605082.014660508
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)
bad3 1+3T 1 + 3T
5 1 1
29 1+29T 1 + 29T
good2 1−3.73T+8T2 1 - 3.73T + 8T^{2}
7 1+11.3T+343T2 1 + 11.3T + 343T^{2}
11 1+55.6T+1.33e3T2 1 + 55.6T + 1.33e3T^{2}
13 1−20.9T+2.19e3T2 1 - 20.9T + 2.19e3T^{2}
17 1−29.8T+4.91e3T2 1 - 29.8T + 4.91e3T^{2}
19 1−25.7T+6.85e3T2 1 - 25.7T + 6.85e3T^{2}
23 1+32.9T+1.21e4T2 1 + 32.9T + 1.21e4T^{2}
31 1+281.T+2.97e4T2 1 + 281.T + 2.97e4T^{2}
37 1−390.T+5.06e4T2 1 - 390.T + 5.06e4T^{2}
41 1+29.4T+6.89e4T2 1 + 29.4T + 6.89e4T^{2}
43 1−9.39T+7.95e4T2 1 - 9.39T + 7.95e4T^{2}
47 1+469.T+1.03e5T2 1 + 469.T + 1.03e5T^{2}
53 1−607.T+1.48e5T2 1 - 607.T + 1.48e5T^{2}
59 1−523.T+2.05e5T2 1 - 523.T + 2.05e5T^{2}
61 1−854.T+2.26e5T2 1 - 854.T + 2.26e5T^{2}
67 1+149.T+3.00e5T2 1 + 149.T + 3.00e5T^{2}
71 1−462.T+3.57e5T2 1 - 462.T + 3.57e5T^{2}
73 1+844.T+3.89e5T2 1 + 844.T + 3.89e5T^{2}
79 1−91.0T+4.93e5T2 1 - 91.0T + 4.93e5T^{2}
83 1+671.T+5.71e5T2 1 + 671.T + 5.71e5T^{2}
89 1+211.T+7.04e5T2 1 + 211.T + 7.04e5T^{2}
97 1−1.77e3T+9.12e5T2 1 - 1.77e3T + 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

−8.667292030565916140117161765266, −7.69531737888007766391260531705, −6.90623280519045130582474639854, −5.97359872202995393617385114455, −5.53435663559863928497869452793, −4.83931138687408654591255118549, −3.85911383208860615814155361991, −3.13881226276043767324723843213, −2.17320708254868037060966983748, −0.51305280189833957909358177687, 0.51305280189833957909358177687, 2.17320708254868037060966983748, 3.13881226276043767324723843213, 3.85911383208860615814155361991, 4.83931138687408654591255118549, 5.53435663559863928497869452793, 5.97359872202995393617385114455, 6.90623280519045130582474639854, 7.69531737888007766391260531705, 8.667292030565916140117161765266

Graph of the ZZ-function along the critical line