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Properties

Label 2-275-1.1-c1-0-14
Degree 22
Conductor 275275
Sign −1-1
Analytic cond. 2.195882.19588
Root an. cond. 1.481851.48185
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.30·2-s − 2.30·3-s − 0.302·4-s − 3·6-s − 0.697·7-s − 3·8-s + 2.30·9-s − 11-s + 0.697·12-s − 5·13-s − 0.908·14-s − 3.30·16-s − 6.90·17-s + 3.00·18-s − 19-s + 1.60·21-s − 1.30·22-s + 7.30·23-s + 6.90·24-s − 6.51·26-s + 1.60·27-s + 0.211·28-s + 0.908·29-s + 10.2·31-s + 1.69·32-s + 2.30·33-s − 9·34-s + ⋯
L(s)  = 1  + 0.921·2-s − 1.32·3-s − 0.151·4-s − 1.22·6-s − 0.263·7-s − 1.06·8-s + 0.767·9-s − 0.301·11-s + 0.201·12-s − 1.38·13-s − 0.242·14-s − 0.825·16-s − 1.67·17-s + 0.707·18-s − 0.229·19-s + 0.350·21-s − 0.277·22-s + 1.52·23-s + 1.41·24-s − 1.27·26-s + 0.308·27-s + 0.0398·28-s + 0.168·29-s + 1.83·31-s + 0.300·32-s + 0.400·33-s − 1.54·34-s + ⋯

Functional equation

Λ(s)=(275s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(275s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 275275    =    52⋅115^{2} \cdot 11
Sign: −1-1
Analytic conductor: 2.195882.19588
Root analytic conductor: 1.481851.48185
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 275, ( :1/2), −1)(2,\ 275,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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)
bad5 1 1
11 1+T 1 + T
good2 1−1.30T+2T2 1 - 1.30T + 2T^{2}
3 1+2.30T+3T2 1 + 2.30T + 3T^{2}
7 1+0.697T+7T2 1 + 0.697T + 7T^{2}
13 1+5T+13T2 1 + 5T + 13T^{2}
17 1+6.90T+17T2 1 + 6.90T + 17T^{2}
19 1+T+19T2 1 + T + 19T^{2}
23 1−7.30T+23T2 1 - 7.30T + 23T^{2}
29 1−0.908T+29T2 1 - 0.908T + 29T^{2}
31 1−10.2T+31T2 1 - 10.2T + 31T^{2}
37 1+2.39T+37T2 1 + 2.39T + 37T^{2}
41 1+5.60T+41T2 1 + 5.60T + 41T^{2}
43 1+7.21T+43T2 1 + 7.21T + 43T^{2}
47 1−3T+47T2 1 - 3T + 47T^{2}
53 1−1.30T+53T2 1 - 1.30T + 53T^{2}
59 1+14.2T+59T2 1 + 14.2T + 59T^{2}
61 1+7.90T+61T2 1 + 7.90T + 61T^{2}
67 1−4T+67T2 1 - 4T + 67T^{2}
71 1+2.60T+71T2 1 + 2.60T + 71T^{2}
73 1−7.90T+73T2 1 - 7.90T + 73T^{2}
79 1+10.9T+79T2 1 + 10.9T + 79T^{2}
83 1−3.51T+83T2 1 - 3.51T + 83T^{2}
89 1−1.69T+89T2 1 - 1.69T + 89T^{2}
97 1+15.3T+97T2 1 + 15.3T + 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

−11.67020501557625141701675509837, −10.75770470419501364913570977747, −9.714380402320605627927132262200, −8.605904675749065979491153211911, −6.92346215499356070623832759682, −6.24236146899732046281076286266, −4.96521826308231203896098221582, −4.66495966213269380614410498206, −2.85320199008313282448555336254, 0, 2.85320199008313282448555336254, 4.66495966213269380614410498206, 4.96521826308231203896098221582, 6.24236146899732046281076286266, 6.92346215499356070623832759682, 8.605904675749065979491153211911, 9.714380402320605627927132262200, 10.75770470419501364913570977747, 11.67020501557625141701675509837

Graph of the ZZ-function along the critical line