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Properties

Label 2-2175-1.1-c1-0-9
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  − 1.14·2-s − 3-s − 0.680·4-s + 1.14·6-s − 3.52·7-s + 3.07·8-s + 9-s + 5.92·11-s + 0.680·12-s − 2.69·13-s + 4.04·14-s − 2.17·16-s + 5.21·17-s − 1.14·18-s − 4.16·19-s + 3.52·21-s − 6.80·22-s − 3.38·23-s − 3.07·24-s + 3.09·26-s − 27-s + 2.39·28-s − 29-s + 1.42·31-s − 3.65·32-s − 5.92·33-s − 5.99·34-s + ⋯
L(s)  = 1  − 0.812·2-s − 0.577·3-s − 0.340·4-s + 0.469·6-s − 1.33·7-s + 1.08·8-s + 0.333·9-s + 1.78·11-s + 0.196·12-s − 0.746·13-s + 1.08·14-s − 0.544·16-s + 1.26·17-s − 0.270·18-s − 0.956·19-s + 0.768·21-s − 1.45·22-s − 0.705·23-s − 0.628·24-s + 0.606·26-s − 0.192·27-s + 0.452·28-s − 0.185·29-s + 0.255·31-s − 0.646·32-s − 1.03·33-s − 1.02·34-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 0.56462554820.5646255482
L(12)L(\frac12) ≈\approx 0.56462554820.5646255482
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+1.14T+2T2 1 + 1.14T + 2T^{2}
7 1+3.52T+7T2 1 + 3.52T + 7T^{2}
11 1−5.92T+11T2 1 - 5.92T + 11T^{2}
13 1+2.69T+13T2 1 + 2.69T + 13T^{2}
17 1−5.21T+17T2 1 - 5.21T + 17T^{2}
19 1+4.16T+19T2 1 + 4.16T + 19T^{2}
23 1+3.38T+23T2 1 + 3.38T + 23T^{2}
31 1−1.42T+31T2 1 - 1.42T + 31T^{2}
37 1+3.36T+37T2 1 + 3.36T + 37T^{2}
41 1−3.78T+41T2 1 - 3.78T + 41T^{2}
43 1+4.17T+43T2 1 + 4.17T + 43T^{2}
47 1+11.3T+47T2 1 + 11.3T + 47T^{2}
53 1−6.14T+53T2 1 - 6.14T + 53T^{2}
59 1+2.59T+59T2 1 + 2.59T + 59T^{2}
61 1+13.0T+61T2 1 + 13.0T + 61T^{2}
67 1−8.72T+67T2 1 - 8.72T + 67T^{2}
71 1−13.8T+71T2 1 - 13.8T + 71T^{2}
73 1−5.58T+73T2 1 - 5.58T + 73T^{2}
79 1−5.07T+79T2 1 - 5.07T + 79T^{2}
83 1−1.68T+83T2 1 - 1.68T + 83T^{2}
89 1−3.16T+89T2 1 - 3.16T + 89T^{2}
97 1−6.72T+97T2 1 - 6.72T + 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

−9.318457848571595702208551918497, −8.432452131819620144158406363708, −7.54675591590105769616006329080, −6.66753181040984061777160331774, −6.21421326042430814936255942425, −5.09993654205724016729633550915, −4.11307252849259335785251141687, −3.41867480863150989055727545481, −1.78161934335717031107631019572, −0.58370932640766198592224533619, 0.58370932640766198592224533619, 1.78161934335717031107631019572, 3.41867480863150989055727545481, 4.11307252849259335785251141687, 5.09993654205724016729633550915, 6.21421326042430814936255942425, 6.66753181040984061777160331774, 7.54675591590105769616006329080, 8.432452131819620144158406363708, 9.318457848571595702208551918497

Graph of the ZZ-function along the critical line