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Properties

Label 2-110670-1.1-c1-0-27
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 − 4·11-s − 12-s − 4·13-s − 14-s − 15-s + 16-s + 17-s − 18-s + 20-s − 21-s + 4·22-s + 24-s + 25-s + 4·26-s − 27-s + 28-s − 2·29-s + 30-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 − 1.20·11-s − 0.288·12-s − 1.10·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.223·20-s − 0.218·21-s + 0.852·22-s + 0.204·24-s + 1/5·25-s + 0.784·26-s − 0.192·27-s + 0.188·28-s − 0.371·29-s + 0.182·30-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+4T+pT2 1 + 4 T + p T^{2}
13 1+4T+pT2 1 + 4 T + p T^{2}
19 1+pT2 1 + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
37 1−4T+pT2 1 - 4 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 1−6T+pT2 1 - 6 T + p T^{2}
53 1−10T+pT2 1 - 10 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+4T+pT2 1 + 4 T + p T^{2}
71 1+2T+pT2 1 + 2 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1−10T+pT2 1 - 10 T + p T^{2}
83 1+4T+pT2 1 + 4 T + p T^{2}
89 1−10T+pT2 1 - 10 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

−13.72858694853988, −13.46986736517781, −12.74717521646160, −12.39635445550610, −11.91153690860612, −11.37724669255844, −10.86871952040400, −10.38878070631201, −10.06054082845393, −9.608604860366874, −9.007875303252846, −8.474158593436810, −7.791603873975314, −7.545519184647879, −6.981721344154608, −6.429806365686981, −5.654905659862866, −5.436733386898711, −4.852913108999670, −4.260017335549170, −3.424704366363025, −2.512192335162444, −2.379367798205579, −1.504066664891541, −0.7405553061935173, 0, 0.7405553061935173, 1.504066664891541, 2.379367798205579, 2.512192335162444, 3.424704366363025, 4.260017335549170, 4.852913108999670, 5.436733386898711, 5.654905659862866, 6.429806365686981, 6.981721344154608, 7.545519184647879, 7.791603873975314, 8.474158593436810, 9.007875303252846, 9.608604860366874, 10.06054082845393, 10.38878070631201, 10.86871952040400, 11.37724669255844, 11.91153690860612, 12.39635445550610, 12.74717521646160, 13.46986736517781, 13.72858694853988

Graph of the ZZ-function along the critical line