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Properties

Label 2-330330-1.1-c1-0-125
Degree 22
Conductor 330330330330
Sign −1-1
Analytic cond. 2637.692637.69
Root an. cond. 51.358551.3585
Motivic weight 11
Arithmetic yes
Rational yes
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 − 5-s − 6-s − 7-s + 8-s + 9-s − 10-s − 12-s − 13-s − 14-s + 15-s + 16-s + 2·17-s + 18-s − 4·19-s − 20-s + 21-s + 4·23-s − 24-s + 25-s − 26-s − 27-s − 28-s + 6·29-s + 30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 0.277·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.917·19-s − 0.223·20-s + 0.218·21-s + 0.834·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.188·28-s + 1.11·29-s + 0.182·30-s + ⋯

Functional equation

Λ(s)=(330330s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 330330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(330330s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 330330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 330330330330    =    2⋅3⋅5⋅7⋅112⋅132 \cdot 3 \cdot 5 \cdot 7 \cdot 11^{2} \cdot 13
Sign: −1-1
Analytic conductor: 2637.692637.69
Root analytic conductor: 51.358551.3585
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 330330, ( :1/2), −1)(2,\ 330330,\ (\ :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+T 1 + T
7 1+T 1 + T
11 1 1
13 1+T 1 + T
good17 1−2T+pT2 1 - 2 T + p T^{2}
19 1+4T+pT2 1 + 4 T + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1−8T+pT2 1 - 8 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1+12T+pT2 1 + 12 T + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1+pT2 1 + p T^{2}
71 1+8T+pT2 1 + 8 T + p T^{2}
73 1+14T+pT2 1 + 14 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1−4T+pT2 1 - 4 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1+10T+pT2 1 + 10 T + p T^{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

−12.77205936151195, −12.28311952506792, −11.95813021341786, −11.75830774348799, −10.82037659147041, −10.77665509433194, −10.34741313331828, −9.682681843873763, −9.277654998742069, −8.550937395902311, −8.275746547784695, −7.543214338976647, −7.163295386783551, −6.723468338647070, −6.286539609957059, −5.697799515020432, −5.397206659239778, −4.611837589060089, −4.406493724345157, −3.894267982237830, −3.169929960494590, −2.768530308464190, −2.212740164037910, −1.317658092853030, −0.8118118533581349, 0, 0.8118118533581349, 1.317658092853030, 2.212740164037910, 2.768530308464190, 3.169929960494590, 3.894267982237830, 4.406493724345157, 4.611837589060089, 5.397206659239778, 5.697799515020432, 6.286539609957059, 6.723468338647070, 7.163295386783551, 7.543214338976647, 8.275746547784695, 8.550937395902311, 9.277654998742069, 9.682681843873763, 10.34741313331828, 10.77665509433194, 10.82037659147041, 11.75830774348799, 11.95813021341786, 12.28311952506792, 12.77205936151195

Graph of the ZZ-function along the critical line