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Properties

Label 2-5610-1.1-c1-0-37
Degree 22
Conductor 56105610
Sign 11
Analytic cond. 44.796044.7960
Root an. cond. 6.692986.69298
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s + 5-s − 6-s + 2·7-s − 8-s + 9-s − 10-s − 11-s + 12-s − 4·13-s − 2·14-s + 15-s + 16-s + 17-s − 18-s + 8·19-s + 20-s + 2·21-s + 22-s − 24-s + 25-s + 4·26-s + 27-s + 2·28-s + 6·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s + 0.288·12-s − 1.10·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 1.83·19-s + 0.223·20-s + 0.436·21-s + 0.213·22-s − 0.204·24-s + 1/5·25-s + 0.784·26-s + 0.192·27-s + 0.377·28-s + 1.11·29-s + ⋯

Functional equation

Λ(s)=(5610s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5610s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 56105610    =    2⋅3⋅5⋅11⋅172 \cdot 3 \cdot 5 \cdot 11 \cdot 17
Sign: 11
Analytic conductor: 44.796044.7960
Root analytic conductor: 6.692986.69298
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 5610, ( :1/2), 1)(2,\ 5610,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.2197265012.219726501
L(12)L(\frac12) ≈\approx 2.2197265012.219726501
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
11 1+T 1 + T
17 1−T 1 - T
good7 1−2T+pT2 1 - 2 T + p T^{2}
13 1+4T+pT2 1 + 4 T + p T^{2}
19 1−8T+pT2 1 - 8 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1−8T+pT2 1 - 8 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1+10T+pT2 1 + 10 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+6T+pT2 1 + 6 T + p T^{2}
61 1−2T+pT2 1 - 2 T + p T^{2}
67 1−2T+pT2 1 - 2 T + p T^{2}
71 1+6T+pT2 1 + 6 T + p T^{2}
73 1−8T+pT2 1 - 8 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1−8T+pT2 1 - 8 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

−8.014577732573335157216093461931, −7.72353265260683119088373234049, −6.93246209178277148027600484537, −6.15247408446868593637176285741, −5.04880624267662423450314672098, −4.76995000226718477111472880364, −3.31819312970869046252227802193, −2.70729266496666967945316346561, −1.81334445828822228623030230293, −0.894276120626598724062637586377, 0.894276120626598724062637586377, 1.81334445828822228623030230293, 2.70729266496666967945316346561, 3.31819312970869046252227802193, 4.76995000226718477111472880364, 5.04880624267662423450314672098, 6.15247408446868593637176285741, 6.93246209178277148027600484537, 7.72353265260683119088373234049, 8.014577732573335157216093461931

Graph of the ZZ-function along the critical line