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Properties

Label 2-8670-1.1-c1-0-169
Degree 22
Conductor 86708670
Sign −1-1
Analytic cond. 69.230269.2302
Root an. cond. 8.320478.32047
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 + 8-s + 9-s − 10-s − 4·11-s + 12-s − 2·13-s − 15-s + 16-s + 18-s + 4·19-s − 20-s − 4·22-s + 24-s + 25-s − 2·26-s + 27-s + 2·29-s − 30-s − 8·31-s + 32-s − 4·33-s + 36-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s + 0.288·12-s − 0.554·13-s − 0.258·15-s + 1/4·16-s + 0.235·18-s + 0.917·19-s − 0.223·20-s − 0.852·22-s + 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.371·29-s − 0.182·30-s − 1.43·31-s + 0.176·32-s − 0.696·33-s + 1/6·36-s + ⋯

Functional equation

Λ(s)=(8670s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(8670s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 86708670    =    2⋅3⋅5⋅1722 \cdot 3 \cdot 5 \cdot 17^{2}
Sign: −1-1
Analytic conductor: 69.230269.2302
Root analytic conductor: 8.320478.32047
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 8670, ( :1/2), −1)(2,\ 8670,\ (\ :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
17 1 1
good7 1+pT2 1 + p T^{2}
11 1+4T+pT2 1 + 4 T + p T^{2}
13 1+2T+pT2 1 + 2 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+8T+pT2 1 + 8 T + p T^{2}
37 1+6T+pT2 1 + 6 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+10T+pT2 1 + 10 T + p T^{2}
59 1+4T+pT2 1 + 4 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+pT2 1 + p T^{2}
73 1−6T+pT2 1 - 6 T + p T^{2}
79 1+8T+pT2 1 + 8 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1−14T+pT2 1 - 14 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

−7.40561162349510633016588055418, −6.91954332593476671184463510324, −5.90592644132209508720459815133, −5.16366564361131865287337583164, −4.71364359659163093237904792022, −3.74770029001862206571218490220, −3.12938112745889251380126144765, −2.47906030853463410883102479535, −1.51008325733353209593744825657, 0, 1.51008325733353209593744825657, 2.47906030853463410883102479535, 3.12938112745889251380126144765, 3.74770029001862206571218490220, 4.71364359659163093237904792022, 5.16366564361131865287337583164, 5.90592644132209508720459815133, 6.91954332593476671184463510324, 7.40561162349510633016588055418

Graph of the ZZ-function along the critical line