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Properties

Label 2-2175-1.1-c1-0-21
Degree 22
Conductor 21752175
Sign 11
Analytic cond. 17.367417.3674
Root an. cond. 4.167424.16742
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  − 2.59·2-s − 3-s + 4.73·4-s + 2.59·6-s + 3.93·7-s − 7.10·8-s + 9-s + 2.24·11-s − 4.73·12-s − 2.09·13-s − 10.2·14-s + 8.97·16-s − 6.95·17-s − 2.59·18-s + 4.29·19-s − 3.93·21-s − 5.83·22-s + 5.19·23-s + 7.10·24-s + 5.43·26-s − 27-s + 18.6·28-s + 29-s + 2.25·31-s − 9.08·32-s − 2.24·33-s + 18.0·34-s + ⋯
L(s)  = 1  − 1.83·2-s − 0.577·3-s + 2.36·4-s + 1.05·6-s + 1.48·7-s − 2.51·8-s + 0.333·9-s + 0.677·11-s − 1.36·12-s − 0.580·13-s − 2.73·14-s + 2.24·16-s − 1.68·17-s − 0.611·18-s + 0.985·19-s − 0.859·21-s − 1.24·22-s + 1.08·23-s + 1.45·24-s + 1.06·26-s − 0.192·27-s + 3.52·28-s + 0.185·29-s + 0.404·31-s − 1.60·32-s − 0.391·33-s + 3.09·34-s + ⋯

Functional equation

Λ(s)=(2175s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2175s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, 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: 17.367417.3674
Root analytic conductor: 4.167424.16742
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2175, ( :1/2), 1)(2,\ 2175,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.74658380920.7465838092
L(12)L(\frac12) ≈\approx 0.74658380920.7465838092
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)
bad3 1+T 1 + T
5 1 1
29 1−T 1 - T
good2 1+2.59T+2T2 1 + 2.59T + 2T^{2}
7 1−3.93T+7T2 1 - 3.93T + 7T^{2}
11 1−2.24T+11T2 1 - 2.24T + 11T^{2}
13 1+2.09T+13T2 1 + 2.09T + 13T^{2}
17 1+6.95T+17T2 1 + 6.95T + 17T^{2}
19 1−4.29T+19T2 1 - 4.29T + 19T^{2}
23 1−5.19T+23T2 1 - 5.19T + 23T^{2}
31 1−2.25T+31T2 1 - 2.25T + 31T^{2}
37 1−10.5T+37T2 1 - 10.5T + 37T^{2}
41 1+2.66T+41T2 1 + 2.66T + 41T^{2}
43 1+6.03T+43T2 1 + 6.03T + 43T^{2}
47 1−7.74T+47T2 1 - 7.74T + 47T^{2}
53 1−1.41T+53T2 1 - 1.41T + 53T^{2}
59 1−5.76T+59T2 1 - 5.76T + 59T^{2}
61 1+10.9T+61T2 1 + 10.9T + 61T^{2}
67 1−11.7T+67T2 1 - 11.7T + 67T^{2}
71 1+4.55T+71T2 1 + 4.55T + 71T^{2}
73 1+12.3T+73T2 1 + 12.3T + 73T^{2}
79 1−14.4T+79T2 1 - 14.4T + 79T^{2}
83 1−3.81T+83T2 1 - 3.81T + 83T^{2}
89 1+2.07T+89T2 1 + 2.07T + 89T^{2}
97 1+12.0T+97T2 1 + 12.0T + 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

−9.054443128455907529195349297512, −8.405350027428219316928162443420, −7.63022892143888121680143117719, −7.02441396769026532923021784992, −6.32094479800245635144797047903, −5.17603007417575509697511783192, −4.35844636948417491593736125408, −2.65670989075033562603567888800, −1.69678880257147556323904755680, −0.807133543343497993372000165724, 0.807133543343497993372000165724, 1.69678880257147556323904755680, 2.65670989075033562603567888800, 4.35844636948417491593736125408, 5.17603007417575509697511783192, 6.32094479800245635144797047903, 7.02441396769026532923021784992, 7.63022892143888121680143117719, 8.405350027428219316928162443420, 9.054443128455907529195349297512

Graph of the ZZ-function along the critical line