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Properties

Label 2-210-1.1-c9-0-30
Degree 22
Conductor 210210
Sign −1-1
Analytic cond. 108.157108.157
Root an. cond. 10.399810.3998
Motivic weight 99
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 16·2-s + 81·3-s + 256·4-s + 625·5-s − 1.29e3·6-s − 2.40e3·7-s − 4.09e3·8-s + 6.56e3·9-s − 1.00e4·10-s + 1.16e4·11-s + 2.07e4·12-s − 5.20e4·13-s + 3.84e4·14-s + 5.06e4·15-s + 6.55e4·16-s − 8.41e4·17-s − 1.04e5·18-s + 4.78e4·19-s + 1.60e5·20-s − 1.94e5·21-s − 1.85e5·22-s + 1.05e6·23-s − 3.31e5·24-s + 3.90e5·25-s + 8.33e5·26-s + 5.31e5·27-s − 6.14e5·28-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 + 0.239·11-s + 0.288·12-s − 0.505·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.244·17-s − 0.235·18-s + 0.0842·19-s + 0.223·20-s − 0.218·21-s − 0.169·22-s + 0.785·23-s − 0.204·24-s + 1/5·25-s + 0.357·26-s + 0.192·27-s − 0.188·28-s + ⋯

Functional equation

Λ(s)=(210s/2 ΓC(s) L(s)=(− Λ(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}
Λ(s)=(210s/2 ΓC(s+9/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 210210    =    2⋅3⋅5⋅72 \cdot 3 \cdot 5 \cdot 7
Sign: −1-1
Analytic conductor: 108.157108.157
Root analytic conductor: 10.399810.3998
Motivic weight: 99
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 210, ( :9/2), −1)(2,\ 210,\ (\ :9/2),\ -1)

Particular Values

L(5)L(5) == 00
L(12)L(\frac12) == 00
L(112)L(\frac{11}{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+p4T 1 + p^{4} T
3 1−p4T 1 - p^{4} T
5 1−p4T 1 - p^{4} T
7 1+p4T 1 + p^{4} T
good11 1−11618T+p9T2 1 - 11618 T + p^{9} T^{2}
13 1+52092T+p9T2 1 + 52092 T + p^{9} T^{2}
17 1+84158T+p9T2 1 + 84158 T + p^{9} T^{2}
19 1−47838T+p9T2 1 - 47838 T + p^{9} T^{2}
23 1−1053540T+p9T2 1 - 1053540 T + p^{9} T^{2}
29 1+7124226T+p9T2 1 + 7124226 T + p^{9} T^{2}
31 1−7102106T+p9T2 1 - 7102106 T + p^{9} T^{2}
37 1+10183078T+p9T2 1 + 10183078 T + p^{9} T^{2}
41 1+22356794T+p9T2 1 + 22356794 T + p^{9} T^{2}
43 1+16360868T+p9T2 1 + 16360868 T + p^{9} T^{2}
47 1−16866016T+p9T2 1 - 16866016 T + p^{9} T^{2}
53 1+32119768T+p9T2 1 + 32119768 T + p^{9} T^{2}
59 1−28695392T+p9T2 1 - 28695392 T + p^{9} T^{2}
61 1+214337930T+p9T2 1 + 214337930 T + p^{9} T^{2}
67 1−311380692T+p9T2 1 - 311380692 T + p^{9} T^{2}
71 1−162400786T+p9T2 1 - 162400786 T + p^{9} T^{2}
73 1−281590340T+p9T2 1 - 281590340 T + p^{9} T^{2}
79 1+594800936T+p9T2 1 + 594800936 T + p^{9} T^{2}
83 1−50037964T+p9T2 1 - 50037964 T + p^{9} T^{2}
89 1−499369010T+p9T2 1 - 499369010 T + p^{9} T^{2}
97 1+680815480T+p9T2 1 + 680815480 T + p^{9} 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

−9.954235539537624247466811853165, −9.314199743999317658312189545943, −8.420117503780066845168621341243, −7.30374888083167657546724294540, −6.44923796625410311911778531786, −5.09813298096857220756058799529, −3.55439824956743029815274090813, −2.45344470484642026009745539868, −1.39586224527908588465691126742, 0, 1.39586224527908588465691126742, 2.45344470484642026009745539868, 3.55439824956743029815274090813, 5.09813298096857220756058799529, 6.44923796625410311911778531786, 7.30374888083167657546724294540, 8.420117503780066845168621341243, 9.314199743999317658312189545943, 9.954235539537624247466811853165

Graph of the ZZ-function along the critical line