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Properties

Label 2-3750-1.1-c1-0-11
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 − 2.92·7-s + 8-s + 9-s + 0.190·11-s − 12-s − 0.310·13-s − 2.92·14-s + 16-s − 6.04·17-s + 18-s + 3.23·19-s + 2.92·21-s + 0.190·22-s − 4.47·23-s − 24-s − 0.310·26-s − 27-s − 2.92·28-s + 6.81·29-s − 1.69·31-s + 32-s − 0.190·33-s − 6.04·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.408·6-s − 1.10·7-s + 0.353·8-s + 0.333·9-s + 0.0573·11-s − 0.288·12-s − 0.0859·13-s − 0.782·14-s + 0.250·16-s − 1.46·17-s + 0.235·18-s + 0.742·19-s + 0.638·21-s + 0.0405·22-s − 0.932·23-s − 0.204·24-s − 0.0608·26-s − 0.192·27-s − 0.553·28-s + 1.26·29-s − 0.303·31-s + 0.176·32-s − 0.0331·33-s − 1.03·34-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 1.9008570951.900857095
L(12)L(\frac12) ≈\approx 1.9008570951.900857095
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+2.92T+7T2 1 + 2.92T + 7T^{2}
11 1−0.190T+11T2 1 - 0.190T + 11T^{2}
13 1+0.310T+13T2 1 + 0.310T + 13T^{2}
17 1+6.04T+17T2 1 + 6.04T + 17T^{2}
19 1−3.23T+19T2 1 - 3.23T + 19T^{2}
23 1+4.47T+23T2 1 + 4.47T + 23T^{2}
29 1−6.81T+29T2 1 - 6.81T + 29T^{2}
31 1+1.69T+31T2 1 + 1.69T + 31T^{2}
37 1−9.75T+37T2 1 - 9.75T + 37T^{2}
41 1−5.07T+41T2 1 - 5.07T + 41T^{2}
43 1−10.2T+43T2 1 - 10.2T + 43T^{2}
47 1−10.7T+47T2 1 - 10.7T + 47T^{2}
53 1+9.97T+53T2 1 + 9.97T + 53T^{2}
59 1−7.42T+59T2 1 - 7.42T + 59T^{2}
61 1+5.54T+61T2 1 + 5.54T + 61T^{2}
67 1−15.0T+67T2 1 - 15.0T + 67T^{2}
71 1−6.85T+71T2 1 - 6.85T + 71T^{2}
73 1+11.6T+73T2 1 + 11.6T + 73T^{2}
79 1−1.57T+79T2 1 - 1.57T + 79T^{2}
83 1+13.0T+83T2 1 + 13.0T + 83T^{2}
89 1+9.28T+89T2 1 + 9.28T + 89T^{2}
97 1+1.73T+97T2 1 + 1.73T + 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

−8.478650524781573919486283580735, −7.48250963862956513260223168254, −6.84566374486949161007670124423, −6.12948579067348350442650719616, −5.72588288672056724278056191451, −4.56013428145610897912324457738, −4.10980112453575596757600651156, −3.02430891941891840023118308040, −2.24210311114820638125546752923, −0.72214120844711079340142348256, 0.72214120844711079340142348256, 2.24210311114820638125546752923, 3.02430891941891840023118308040, 4.10980112453575596757600651156, 4.56013428145610897912324457738, 5.72588288672056724278056191451, 6.12948579067348350442650719616, 6.84566374486949161007670124423, 7.48250963862956513260223168254, 8.478650524781573919486283580735

Graph of the ZZ-function along the critical line