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Properties

Label 2-3750-1.1-c1-0-23
Degree 22
Conductor 37503750
Sign 11
Analytic cond. 29.943929.9439
Root an. cond. 5.472105.47210
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-s − 3-s + 4-s + 6-s + 3.52·7-s − 8-s + 9-s + 5.25·11-s − 12-s − 0.619·13-s − 3.52·14-s + 16-s − 3.44·17-s − 18-s + 2.27·19-s − 3.52·21-s − 5.25·22-s + 8.95·23-s + 24-s + 0.619·26-s − 27-s + 3.52·28-s − 2.68·29-s + 9.76·31-s − 32-s − 5.25·33-s + 3.44·34-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s + 1.33·7-s − 0.353·8-s + 0.333·9-s + 1.58·11-s − 0.288·12-s − 0.171·13-s − 0.941·14-s + 0.250·16-s − 0.835·17-s − 0.235·18-s + 0.521·19-s − 0.768·21-s − 1.12·22-s + 1.86·23-s + 0.204·24-s + 0.121·26-s − 0.192·27-s + 0.665·28-s − 0.497·29-s + 1.75·31-s − 0.176·32-s − 0.914·33-s + 0.590·34-s + ⋯

Functional equation

Λ(s)=(3750s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3750s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 37503750    =    2⋅3⋅542 \cdot 3 \cdot 5^{4}
Sign: 11
Analytic conductor: 29.943929.9439
Root analytic conductor: 5.472105.47210
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3750, ( :1/2), 1)(2,\ 3750,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.5867669391.586766939
L(12)L(\frac12) ≈\approx 1.5867669391.586766939
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)
bad2 1+T 1 + T
3 1+T 1 + T
5 1 1
good7 1−3.52T+7T2 1 - 3.52T + 7T^{2}
11 1−5.25T+11T2 1 - 5.25T + 11T^{2}
13 1+0.619T+13T2 1 + 0.619T + 13T^{2}
17 1+3.44T+17T2 1 + 3.44T + 17T^{2}
19 1−2.27T+19T2 1 - 2.27T + 19T^{2}
23 1−8.95T+23T2 1 - 8.95T + 23T^{2}
29 1+2.68T+29T2 1 + 2.68T + 29T^{2}
31 1−9.76T+31T2 1 - 9.76T + 31T^{2}
37 1−1.00T+37T2 1 - 1.00T + 37T^{2}
41 1−6.29T+41T2 1 - 6.29T + 41T^{2}
43 1−1.51T+43T2 1 - 1.51T + 43T^{2}
47 1+10.6T+47T2 1 + 10.6T + 47T^{2}
53 1+0.553T+53T2 1 + 0.553T + 53T^{2}
59 1−0.278T+59T2 1 - 0.278T + 59T^{2}
61 1−3.68T+61T2 1 - 3.68T + 61T^{2}
67 1−11.9T+67T2 1 - 11.9T + 67T^{2}
71 1−4.74T+71T2 1 - 4.74T + 71T^{2}
73 1+9.83T+73T2 1 + 9.83T + 73T^{2}
79 1−6.52T+79T2 1 - 6.52T + 79T^{2}
83 1+2.65T+83T2 1 + 2.65T + 83T^{2}
89 1+13.3T+89T2 1 + 13.3T + 89T^{2}
97 1+4.37T+97T2 1 + 4.37T + 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

−8.502914681103929140112778135209, −7.88767740145923799787626344778, −6.88877324470737021411325217404, −6.62595342699649202641893658830, −5.52059152220800558485862716221, −4.75670985409453096161123168318, −4.06229043309977308203186308105, −2.77160724666157675633142735290, −1.58064045692853118509980946767, −0.943408881415150737903748193077, 0.943408881415150737903748193077, 1.58064045692853118509980946767, 2.77160724666157675633142735290, 4.06229043309977308203186308105, 4.75670985409453096161123168318, 5.52059152220800558485862716221, 6.62595342699649202641893658830, 6.88877324470737021411325217404, 7.88767740145923799787626344778, 8.502914681103929140112778135209

Graph of the ZZ-function along the critical line