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Properties

Label 2-2175-1.1-c1-0-31
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  − 2.47·2-s + 3-s + 4.14·4-s − 2.47·6-s − 2.83·7-s − 5.30·8-s + 9-s + 6.09·11-s + 4.14·12-s + 3.00·13-s + 7.02·14-s + 4.86·16-s + 2.74·17-s − 2.47·18-s + 8.03·19-s − 2.83·21-s − 15.1·22-s + 7.56·23-s − 5.30·24-s − 7.43·26-s + 27-s − 11.7·28-s − 29-s − 5.32·31-s − 1.44·32-s + 6.09·33-s − 6.79·34-s + ⋯
L(s)  = 1  − 1.75·2-s + 0.577·3-s + 2.07·4-s − 1.01·6-s − 1.07·7-s − 1.87·8-s + 0.333·9-s + 1.83·11-s + 1.19·12-s + 0.832·13-s + 1.87·14-s + 1.21·16-s + 0.664·17-s − 0.584·18-s + 1.84·19-s − 0.618·21-s − 3.22·22-s + 1.57·23-s − 1.08·24-s − 1.45·26-s + 0.192·27-s − 2.21·28-s − 0.185·29-s − 0.955·31-s − 0.255·32-s + 1.06·33-s − 1.16·34-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 1.0828121351.082812135
L(12)L(\frac12) ≈\approx 1.0828121351.082812135
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+2.47T+2T2 1 + 2.47T + 2T^{2}
7 1+2.83T+7T2 1 + 2.83T + 7T^{2}
11 1−6.09T+11T2 1 - 6.09T + 11T^{2}
13 1−3.00T+13T2 1 - 3.00T + 13T^{2}
17 1−2.74T+17T2 1 - 2.74T + 17T^{2}
19 1−8.03T+19T2 1 - 8.03T + 19T^{2}
23 1−7.56T+23T2 1 - 7.56T + 23T^{2}
31 1+5.32T+31T2 1 + 5.32T + 31T^{2}
37 1+6.28T+37T2 1 + 6.28T + 37T^{2}
41 1+9.04T+41T2 1 + 9.04T + 41T^{2}
43 1+9.06T+43T2 1 + 9.06T + 43T^{2}
47 1−0.0839T+47T2 1 - 0.0839T + 47T^{2}
53 1+3.82T+53T2 1 + 3.82T + 53T^{2}
59 1−11.9T+59T2 1 - 11.9T + 59T^{2}
61 1+0.275T+61T2 1 + 0.275T + 61T^{2}
67 1+7.36T+67T2 1 + 7.36T + 67T^{2}
71 1−15.2T+71T2 1 - 15.2T + 71T^{2}
73 1−8.77T+73T2 1 - 8.77T + 73T^{2}
79 1−1.64T+79T2 1 - 1.64T + 79T^{2}
83 1−1.38T+83T2 1 - 1.38T + 83T^{2}
89 1+1.05T+89T2 1 + 1.05T + 89T^{2}
97 1−9.18T+97T2 1 - 9.18T + 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

−9.046516776942070549927265557045, −8.649888309574938671710872052215, −7.64722829957188565052078113897, −6.82302620325241651382512794166, −6.59910361158590620883203235904, −5.31683611640129005011865889786, −3.50368996007566118267747892512, −3.28802163077325270752652256117, −1.68332697349680870697875235883, −0.939099081887040149236553067373, 0.939099081887040149236553067373, 1.68332697349680870697875235883, 3.28802163077325270752652256117, 3.50368996007566118267747892512, 5.31683611640129005011865889786, 6.59910361158590620883203235904, 6.82302620325241651382512794166, 7.64722829957188565052078113897, 8.649888309574938671710872052215, 9.046516776942070549927265557045

Graph of the ZZ-function along the critical line