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Properties

Label 2-110670-1.1-c1-0-52
Degree 22
Conductor 110670110670
Sign −1-1
Analytic cond. 883.704883.704
Root an. cond. 29.727129.7271
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 − 2·11-s − 12-s − 14-s + 15-s + 16-s + 17-s − 18-s + 6·19-s − 20-s − 21-s + 2·22-s + 4·23-s + 24-s + 25-s − 27-s + 28-s − 30-s − 31-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 − 0.603·11-s − 0.288·12-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 1.37·19-s − 0.223·20-s − 0.218·21-s + 0.426·22-s + 0.834·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s + 0.188·28-s − 0.182·30-s − 0.179·31-s + ⋯

Functional equation

Λ(s)=(110670s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 110670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(110670s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 110670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 110670110670    =    2⋅3⋅5⋅7⋅17⋅312 \cdot 3 \cdot 5 \cdot 7 \cdot 17 \cdot 31
Sign: −1-1
Analytic conductor: 883.704883.704
Root analytic conductor: 29.727129.7271
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 110670, ( :1/2), −1)(2,\ 110670,\ (\ :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
17 1−T 1 - T
31 1+T 1 + T
good11 1+2T+pT2 1 + 2 T + p T^{2}
13 1+pT2 1 + p T^{2}
19 1−6T+pT2 1 - 6 T + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1+pT2 1 + p T^{2}
37 1−8T+pT2 1 - 8 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1+pT2 1 + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+2T+pT2 1 + 2 T + p T^{2}
59 1−6T+pT2 1 - 6 T + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1−8T+pT2 1 - 8 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1−10T+pT2 1 - 10 T + p T^{2}
79 1−4T+pT2 1 - 4 T + p T^{2}
83 1−10T+pT2 1 - 10 T + p T^{2}
89 1−14T+pT2 1 - 14 T + p T^{2}
97 1+18T+pT2 1 + 18 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

−13.85883039644091, −13.34982896827936, −12.73709909578248, −12.40990967803345, −11.69897223450593, −11.47688650305811, −10.91163945286522, −10.65048250906209, −9.895367537192599, −9.524876259879174, −9.088798987657410, −8.314622718226372, −7.863262740351256, −7.589133446997568, −6.949517566053060, −6.506593419776799, −5.780973746491806, −5.166353474579439, −5.006768273024971, −4.022506436923724, −3.585166957781820, −2.703043867331787, −2.337091080874357, −1.164441260877578, −0.9636633658183337, 0, 0.9636633658183337, 1.164441260877578, 2.337091080874357, 2.703043867331787, 3.585166957781820, 4.022506436923724, 5.006768273024971, 5.166353474579439, 5.780973746491806, 6.506593419776799, 6.949517566053060, 7.589133446997568, 7.863262740351256, 8.314622718226372, 9.088798987657410, 9.524876259879174, 9.895367537192599, 10.65048250906209, 10.91163945286522, 11.47688650305811, 11.69897223450593, 12.40990967803345, 12.73709909578248, 13.34982896827936, 13.85883039644091

Graph of the ZZ-function along the critical line