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Properties

Label 2-2175-1.1-c1-0-4
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  + 0.135·2-s − 3-s − 1.98·4-s − 0.135·6-s − 1.12·7-s − 0.537·8-s + 9-s − 5.08·11-s + 1.98·12-s + 0.338·13-s − 0.151·14-s + 3.89·16-s − 1.77·17-s + 0.135·18-s − 5.13·19-s + 1.12·21-s − 0.685·22-s − 7.31·23-s + 0.537·24-s + 0.0457·26-s − 27-s + 2.22·28-s + 29-s + 8.74·31-s + 1.60·32-s + 5.08·33-s − 0.238·34-s + ⋯
L(s)  = 1  + 0.0954·2-s − 0.577·3-s − 0.990·4-s − 0.0551·6-s − 0.424·7-s − 0.190·8-s + 0.333·9-s − 1.53·11-s + 0.572·12-s + 0.0939·13-s − 0.0405·14-s + 0.972·16-s − 0.429·17-s + 0.0318·18-s − 1.17·19-s + 0.245·21-s − 0.146·22-s − 1.52·23-s + 0.109·24-s + 0.00897·26-s − 0.192·27-s + 0.420·28-s + 0.185·29-s + 1.57·31-s + 0.282·32-s + 0.884·33-s − 0.0409·34-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 0.55516991180.5551699118
L(12)L(\frac12) ≈\approx 0.55516991180.5551699118
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 1−0.135T+2T2 1 - 0.135T + 2T^{2}
7 1+1.12T+7T2 1 + 1.12T + 7T^{2}
11 1+5.08T+11T2 1 + 5.08T + 11T^{2}
13 1−0.338T+13T2 1 - 0.338T + 13T^{2}
17 1+1.77T+17T2 1 + 1.77T + 17T^{2}
19 1+5.13T+19T2 1 + 5.13T + 19T^{2}
23 1+7.31T+23T2 1 + 7.31T + 23T^{2}
31 1−8.74T+31T2 1 - 8.74T + 31T^{2}
37 1−9.40T+37T2 1 - 9.40T + 37T^{2}
41 1−8.95T+41T2 1 - 8.95T + 41T^{2}
43 1+12.3T+43T2 1 + 12.3T + 43T^{2}
47 1−2.12T+47T2 1 - 2.12T + 47T^{2}
53 1+10.9T+53T2 1 + 10.9T + 53T^{2}
59 1+9.22T+59T2 1 + 9.22T + 59T^{2}
61 1−1.48T+61T2 1 - 1.48T + 61T^{2}
67 1+9.41T+67T2 1 + 9.41T + 67T^{2}
71 1−11.1T+71T2 1 - 11.1T + 71T^{2}
73 1−2.51T+73T2 1 - 2.51T + 73T^{2}
79 1−11.2T+79T2 1 - 11.2T + 79T^{2}
83 1−11.4T+83T2 1 - 11.4T + 83T^{2}
89 1−16.0T+89T2 1 - 16.0T + 89T^{2}
97 1−9.76T+97T2 1 - 9.76T + 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

−9.147530676922451589239639638888, −8.050416247444358702929420850645, −7.894379646578534421335950158219, −6.33373242326313554291353086860, −6.05790764408037266222091762969, −4.85179999782983522438846966953, −4.50919853439583452482543824739, −3.36305665104416248127961615640, −2.22533892996369667379036589032, −0.47509649146166903336571070911, 0.47509649146166903336571070911, 2.22533892996369667379036589032, 3.36305665104416248127961615640, 4.50919853439583452482543824739, 4.85179999782983522438846966953, 6.05790764408037266222091762969, 6.33373242326313554291353086860, 7.894379646578534421335950158219, 8.050416247444358702929420850645, 9.147530676922451589239639638888

Graph of the ZZ-function along the critical line