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Properties

Label 2-2175-1.1-c1-0-34
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  + 1.98·2-s − 3-s + 1.93·4-s − 1.98·6-s − 2.16·7-s − 0.119·8-s + 9-s + 3.44·11-s − 1.93·12-s + 3.74·13-s − 4.29·14-s − 4.11·16-s − 4.33·17-s + 1.98·18-s + 3.90·19-s + 2.16·21-s + 6.84·22-s + 3.78·23-s + 0.119·24-s + 7.42·26-s − 27-s − 4.20·28-s + 29-s + 10.3·31-s − 7.93·32-s − 3.44·33-s − 8.60·34-s + ⋯
L(s)  = 1  + 1.40·2-s − 0.577·3-s + 0.969·4-s − 0.810·6-s − 0.818·7-s − 0.0423·8-s + 0.333·9-s + 1.03·11-s − 0.559·12-s + 1.03·13-s − 1.14·14-s − 1.02·16-s − 1.05·17-s + 0.467·18-s + 0.895·19-s + 0.472·21-s + 1.45·22-s + 0.789·23-s + 0.0244·24-s + 1.45·26-s − 0.192·27-s − 0.794·28-s + 0.185·29-s + 1.85·31-s − 1.40·32-s − 0.600·33-s − 1.47·34-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 3.0038365223.003836522
L(12)L(\frac12) ≈\approx 3.0038365223.003836522
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 1−1.98T+2T2 1 - 1.98T + 2T^{2}
7 1+2.16T+7T2 1 + 2.16T + 7T^{2}
11 1−3.44T+11T2 1 - 3.44T + 11T^{2}
13 1−3.74T+13T2 1 - 3.74T + 13T^{2}
17 1+4.33T+17T2 1 + 4.33T + 17T^{2}
19 1−3.90T+19T2 1 - 3.90T + 19T^{2}
23 1−3.78T+23T2 1 - 3.78T + 23T^{2}
31 1−10.3T+31T2 1 - 10.3T + 31T^{2}
37 1−7.70T+37T2 1 - 7.70T + 37T^{2}
41 1−7.88T+41T2 1 - 7.88T + 41T^{2}
43 1+0.975T+43T2 1 + 0.975T + 43T^{2}
47 1+12.1T+47T2 1 + 12.1T + 47T^{2}
53 1−13.1T+53T2 1 - 13.1T + 53T^{2}
59 1−1.47T+59T2 1 - 1.47T + 59T^{2}
61 1+4.24T+61T2 1 + 4.24T + 61T^{2}
67 1−6.74T+67T2 1 - 6.74T + 67T^{2}
71 1−10.3T+71T2 1 - 10.3T + 71T^{2}
73 1−12.3T+73T2 1 - 12.3T + 73T^{2}
79 1+5.02T+79T2 1 + 5.02T + 79T^{2}
83 1+10.4T+83T2 1 + 10.4T + 83T^{2}
89 1−4.35T+89T2 1 - 4.35T + 89T^{2}
97 1+3.75T+97T2 1 + 3.75T + 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

−9.218169936895908471074162013214, −8.287617189628156252583816964597, −6.92134943470379853560250713443, −6.46239491474346456039077817011, −5.97702098799257871085606062883, −4.98247156281963945267549344676, −4.22984552967165552012042898578, −3.52534545507026369945491233549, −2.60968081231717799704858326995, −0.983878420232591295657335447188, 0.983878420232591295657335447188, 2.60968081231717799704858326995, 3.52534545507026369945491233549, 4.22984552967165552012042898578, 4.98247156281963945267549344676, 5.97702098799257871085606062883, 6.46239491474346456039077817011, 6.92134943470379853560250713443, 8.287617189628156252583816964597, 9.218169936895908471074162013214

Graph of the ZZ-function along the critical line