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Properties

Label 2-3750-1.1-c1-0-48
Degree 22
Conductor 37503750
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 6-s + 2·7-s − 8-s + 9-s − 5.23·11-s − 12-s − 4.85·13-s − 2·14-s + 16-s + 7.85·17-s − 18-s − 2.76·19-s − 2·21-s + 5.23·22-s + 6·23-s + 24-s + 4.85·26-s − 27-s + 2·28-s − 1.38·29-s + 3.70·31-s − 32-s + 5.23·33-s − 7.85·34-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 0.333·9-s − 1.57·11-s − 0.288·12-s − 1.34·13-s − 0.534·14-s + 0.250·16-s + 1.90·17-s − 0.235·18-s − 0.634·19-s − 0.436·21-s + 1.11·22-s + 1.25·23-s + 0.204·24-s + 0.951·26-s − 0.192·27-s + 0.377·28-s − 0.256·29-s + 0.666·31-s − 0.176·32-s + 0.911·33-s − 1.34·34-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: −1-1
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: 11
Selberg data: (2, 3750, ( :1/2), −1)(2,\ 3750,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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 1−2T+7T2 1 - 2T + 7T^{2}
11 1+5.23T+11T2 1 + 5.23T + 11T^{2}
13 1+4.85T+13T2 1 + 4.85T + 13T^{2}
17 1−7.85T+17T2 1 - 7.85T + 17T^{2}
19 1+2.76T+19T2 1 + 2.76T + 19T^{2}
23 1−6T+23T2 1 - 6T + 23T^{2}
29 1+1.38T+29T2 1 + 1.38T + 29T^{2}
31 1−3.70T+31T2 1 - 3.70T + 31T^{2}
37 1+2.14T+37T2 1 + 2.14T + 37T^{2}
41 1+6.09T+41T2 1 + 6.09T + 41T^{2}
43 1+1.23T+43T2 1 + 1.23T + 43T^{2}
47 1−4.76T+47T2 1 - 4.76T + 47T^{2}
53 1−8.56T+53T2 1 - 8.56T + 53T^{2}
59 1−8.94T+59T2 1 - 8.94T + 59T^{2}
61 1+8.85T+61T2 1 + 8.85T + 61T^{2}
67 1+9.70T+67T2 1 + 9.70T + 67T^{2}
71 1+14.1T+71T2 1 + 14.1T + 71T^{2}
73 1+3.14T+73T2 1 + 3.14T + 73T^{2}
79 1+79T2 1 + 79T^{2}
83 1−6T+83T2 1 - 6T + 83T^{2}
89 1−1.38T+89T2 1 - 1.38T + 89T^{2}
97 1+13.8T+97T2 1 + 13.8T + 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

−7.997055380776331572648735866953, −7.52956833677045865887194948170, −6.95334655871307471255067266908, −5.74751437804703587918797517088, −5.22939497959559114773746464413, −4.61914620406100735710287348611, −3.17617963695677721852284600107, −2.37525957408376665510133714282, −1.23701953300911402566996408412, 0, 1.23701953300911402566996408412, 2.37525957408376665510133714282, 3.17617963695677721852284600107, 4.61914620406100735710287348611, 5.22939497959559114773746464413, 5.74751437804703587918797517088, 6.95334655871307471255067266908, 7.52956833677045865887194948170, 7.997055380776331572648735866953

Graph of the ZZ-function along the critical line