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Properties

Label 2-3330-1.1-c1-0-53
Degree 22
Conductor 33303330
Sign −1-1
Analytic cond. 26.590126.5901
Root an. cond. 5.156565.15656
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 + 4-s − 5-s − 3.37·7-s + 8-s − 10-s + 1.37·11-s + 1.37·13-s − 3.37·14-s + 16-s + 1.37·17-s − 1.37·19-s − 20-s + 1.37·22-s − 3.37·23-s + 25-s + 1.37·26-s − 3.37·28-s − 6·29-s + 2.74·31-s + 32-s + 1.37·34-s + 3.37·35-s − 37-s − 1.37·38-s − 40-s − 8.74·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 0.447·5-s − 1.27·7-s + 0.353·8-s − 0.316·10-s + 0.413·11-s + 0.380·13-s − 0.901·14-s + 0.250·16-s + 0.332·17-s − 0.314·19-s − 0.223·20-s + 0.292·22-s − 0.703·23-s + 0.200·25-s + 0.269·26-s − 0.637·28-s − 1.11·29-s + 0.492·31-s + 0.176·32-s + 0.235·34-s + 0.570·35-s − 0.164·37-s − 0.222·38-s − 0.158·40-s − 1.36·41-s + ⋯

Functional equation

Λ(s)=(3330s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(3330s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33303330    =    2⋅32⋅5⋅372 \cdot 3^{2} \cdot 5 \cdot 37
Sign: −1-1
Analytic conductor: 26.590126.5901
Root analytic conductor: 5.156565.15656
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 3330, ( :1/2), −1)(2,\ 3330,\ (\ :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 1
5 1+T 1 + T
37 1+T 1 + T
good7 1+3.37T+7T2 1 + 3.37T + 7T^{2}
11 1−1.37T+11T2 1 - 1.37T + 11T^{2}
13 1−1.37T+13T2 1 - 1.37T + 13T^{2}
17 1−1.37T+17T2 1 - 1.37T + 17T^{2}
19 1+1.37T+19T2 1 + 1.37T + 19T^{2}
23 1+3.37T+23T2 1 + 3.37T + 23T^{2}
29 1+6T+29T2 1 + 6T + 29T^{2}
31 1−2.74T+31T2 1 - 2.74T + 31T^{2}
41 1+8.74T+41T2 1 + 8.74T + 41T^{2}
43 1+4T+43T2 1 + 4T + 43T^{2}
47 1−4.74T+47T2 1 - 4.74T + 47T^{2}
53 1+5.37T+53T2 1 + 5.37T + 53T^{2}
59 1+14.7T+59T2 1 + 14.7T + 59T^{2}
61 1+2.74T+61T2 1 + 2.74T + 61T^{2}
67 1−2.74T+67T2 1 - 2.74T + 67T^{2}
71 1+1.25T+71T2 1 + 1.25T + 71T^{2}
73 1+4.11T+73T2 1 + 4.11T + 73T^{2}
79 1−4T+79T2 1 - 4T + 79T^{2}
83 1+0.627T+83T2 1 + 0.627T + 83T^{2}
89 1+13.3T+89T2 1 + 13.3T + 89T^{2}
97 1−13.4T+97T2 1 - 13.4T + 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

−8.151112827566941020153314655912, −7.36716726103839439926041750527, −6.54859484058128516811337610090, −6.12314964877450324636714959788, −5.20782266792309859734974093443, −4.18185752622285749252271334572, −3.56666661532034164746079117739, −2.88675142955439155506145127817, −1.60753780957048935469755045442, 0, 1.60753780957048935469755045442, 2.88675142955439155506145127817, 3.56666661532034164746079117739, 4.18185752622285749252271334572, 5.20782266792309859734974093443, 6.12314964877450324636714959788, 6.54859484058128516811337610090, 7.36716726103839439926041750527, 8.151112827566941020153314655912

Graph of the ZZ-function along the critical line