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Properties

Label 2-60840-1.1-c1-0-41
Degree 22
Conductor 6084060840
Sign −1-1
Analytic cond. 485.809485.809
Root an. cond. 22.041022.0410
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  − 5-s + 6·17-s − 4·19-s + 25-s + 2·29-s + 2·37-s − 2·41-s − 4·43-s + 4·47-s − 7·49-s + 10·53-s + 8·59-s − 2·61-s + 4·67-s − 12·71-s + 6·73-s − 16·83-s − 6·85-s − 10·89-s + 4·95-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.45·17-s − 0.917·19-s + 1/5·25-s + 0.371·29-s + 0.328·37-s − 0.312·41-s − 0.609·43-s + 0.583·47-s − 49-s + 1.37·53-s + 1.04·59-s − 0.256·61-s + 0.488·67-s − 1.42·71-s + 0.702·73-s − 1.75·83-s − 0.650·85-s − 1.05·89-s + 0.410·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

Λ(s)=(60840s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 60840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(60840s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 60840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 6084060840    =    23⋅32⋅5⋅1322^{3} \cdot 3^{2} \cdot 5 \cdot 13^{2}
Sign: −1-1
Analytic conductor: 485.809485.809
Root analytic conductor: 22.041022.0410
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 60840, ( :1/2), −1)(2,\ 60840,\ (\ :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 1
3 1 1
5 1+T 1 + T
13 1 1
good7 1+pT2 1 + p T^{2}
11 1+pT2 1 + p T^{2}
17 1−6T+pT2 1 - 6 T + p T^{2}
19 1+4T+pT2 1 + 4 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1−2T+pT2 1 - 2 T + p T^{2}
31 1+pT2 1 + 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 1−4T+pT2 1 - 4 T + p T^{2}
53 1−10T+pT2 1 - 10 T + p T^{2}
59 1−8T+pT2 1 - 8 T + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1+12T+pT2 1 + 12 T + p T^{2}
73 1−6T+pT2 1 - 6 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1+16T+pT2 1 + 16 T + p T^{2}
89 1+10T+pT2 1 + 10 T + p T^{2}
97 1+2T+pT2 1 + 2 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

−14.62653301522286, −14.13069571574099, −13.51851560717663, −12.98600004425497, −12.48868052295855, −12.05276296482323, −11.54044367732055, −11.06968900957357, −10.36315149274344, −10.05362229402391, −9.513388448904866, −8.695224999738117, −8.414237516919908, −7.828819709240526, −7.276443369981650, −6.754477748442332, −6.121691110719434, −5.515535215177466, −5.014961039267487, −4.237643016474681, −3.822313632434255, −3.093905360495723, −2.536049927611069, −1.636274357960329, −0.9318376170380364, 0, 0.9318376170380364, 1.636274357960329, 2.536049927611069, 3.093905360495723, 3.822313632434255, 4.237643016474681, 5.014961039267487, 5.515535215177466, 6.121691110719434, 6.754477748442332, 7.276443369981650, 7.828819709240526, 8.414237516919908, 8.695224999738117, 9.513388448904866, 10.05362229402391, 10.36315149274344, 11.06968900957357, 11.54044367732055, 12.05276296482323, 12.48868052295855, 12.98600004425497, 13.51851560717663, 14.13069571574099, 14.62653301522286

Graph of the ZZ-function along the critical line