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Properties

Label 2-3750-1.1-c1-0-35
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 + 0.929·7-s + 8-s + 9-s − 1.29·11-s + 12-s + 0.581·13-s + 0.929·14-s + 16-s + 0.338·17-s + 18-s − 1.27·19-s + 0.929·21-s − 1.29·22-s + 0.403·23-s + 24-s + 0.581·26-s + 27-s + 0.929·28-s + 8.82·29-s + 5.29·31-s + 32-s − 1.29·33-s + 0.338·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.408·6-s + 0.351·7-s + 0.353·8-s + 0.333·9-s − 0.390·11-s + 0.288·12-s + 0.161·13-s + 0.248·14-s + 0.250·16-s + 0.0820·17-s + 0.235·18-s − 0.293·19-s + 0.202·21-s − 0.276·22-s + 0.0841·23-s + 0.204·24-s + 0.114·26-s + 0.192·27-s + 0.175·28-s + 1.63·29-s + 0.950·31-s + 0.176·32-s − 0.225·33-s + 0.0580·34-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 4.0808776734.080877673
L(12)L(\frac12) ≈\approx 4.0808776734.080877673
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 1−0.929T+7T2 1 - 0.929T + 7T^{2}
11 1+1.29T+11T2 1 + 1.29T + 11T^{2}
13 1−0.581T+13T2 1 - 0.581T + 13T^{2}
17 1−0.338T+17T2 1 - 0.338T + 17T^{2}
19 1+1.27T+19T2 1 + 1.27T + 19T^{2}
23 1−0.403T+23T2 1 - 0.403T + 23T^{2}
29 1−8.82T+29T2 1 - 8.82T + 29T^{2}
31 1−5.29T+31T2 1 - 5.29T + 31T^{2}
37 1−3.51T+37T2 1 - 3.51T + 37T^{2}
41 1−3.05T+41T2 1 - 3.05T + 41T^{2}
43 1−4.16T+43T2 1 - 4.16T + 43T^{2}
47 1+7.68T+47T2 1 + 7.68T + 47T^{2}
53 1−11.5T+53T2 1 - 11.5T + 53T^{2}
59 1−7.83T+59T2 1 - 7.83T + 59T^{2}
61 1+13.9T+61T2 1 + 13.9T + 61T^{2}
67 1−7.97T+67T2 1 - 7.97T + 67T^{2}
71 1−8.54T+71T2 1 - 8.54T + 71T^{2}
73 1−3.68T+73T2 1 - 3.68T + 73T^{2}
79 1−14.7T+79T2 1 - 14.7T + 79T^{2}
83 1−2.45T+83T2 1 - 2.45T + 83T^{2}
89 1+8.97T+89T2 1 + 8.97T + 89T^{2}
97 1+18.9T+97T2 1 + 18.9T + 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

−8.171480466878935991491583689244, −8.015040775045239728259798156620, −6.89471796549311376899027549986, −6.35439666086411747990446097365, −5.36138301624236024092494538186, −4.64881594726560737127305854195, −3.94120912820747692065226998192, −2.94841463796701756914533878385, −2.30711202182688657394638661980, −1.08884274022129004986707438654, 1.08884274022129004986707438654, 2.30711202182688657394638661980, 2.94841463796701756914533878385, 3.94120912820747692065226998192, 4.64881594726560737127305854195, 5.36138301624236024092494538186, 6.35439666086411747990446097365, 6.89471796549311376899027549986, 8.015040775045239728259798156620, 8.171480466878935991491583689244

Graph of the ZZ-function along the critical line