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Properties

Label 2-605-1.1-c1-0-21
Degree 22
Conductor 605605
Sign −1-1
Analytic cond. 4.830944.83094
Root an. cond. 2.197942.19794
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 − 4-s + 5-s + 3·8-s − 3·9-s − 10-s − 2·13-s − 16-s − 6·17-s + 3·18-s + 4·19-s − 20-s + 4·23-s + 25-s + 2·26-s − 6·29-s − 8·31-s − 5·32-s + 6·34-s + 3·36-s − 2·37-s − 4·38-s + 3·40-s − 2·41-s − 4·43-s − 3·45-s − 4·46-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.447·5-s + 1.06·8-s − 9-s − 0.316·10-s − 0.554·13-s − 1/4·16-s − 1.45·17-s + 0.707·18-s + 0.917·19-s − 0.223·20-s + 0.834·23-s + 1/5·25-s + 0.392·26-s − 1.11·29-s − 1.43·31-s − 0.883·32-s + 1.02·34-s + 1/2·36-s − 0.328·37-s − 0.648·38-s + 0.474·40-s − 0.312·41-s − 0.609·43-s − 0.447·45-s − 0.589·46-s + ⋯

Functional equation

Λ(s)=(605s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(605s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 605605    =    5⋅1125 \cdot 11^{2}
Sign: −1-1
Analytic conductor: 4.830944.83094
Root analytic conductor: 2.197942.19794
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 605, ( :1/2), −1)(2,\ 605,\ (\ :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)
bad5 1−T 1 - T
11 1 1
good2 1+T+pT2 1 + T + p T^{2}
3 1+pT2 1 + p T^{2}
7 1+pT2 1 + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1−4T+pT2 1 - 4 T + 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+2T+pT2 1 + 2 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1+12T+pT2 1 + 12 T + p T^{2}
53 1+2T+pT2 1 + 2 T + p T^{2}
59 1−4T+pT2 1 - 4 T + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1+16T+pT2 1 + 16 T + p T^{2}
71 1−8T+pT2 1 - 8 T + p T^{2}
73 1+14T+pT2 1 + 14 T + p T^{2}
79 1+8T+pT2 1 + 8 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 1−10T+pT2 1 - 10 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

−10.02041446476298683486486191886, −9.196982768972178400591968116867, −8.758832935924856173060099057787, −7.71367077872044547263533331174, −6.79405906039010685182823940446, −5.50821445395345299948641768104, −4.77278357968932846404359363084, −3.32452113574721608186627592149, −1.86853886263346253357248352832, 0, 1.86853886263346253357248352832, 3.32452113574721608186627592149, 4.77278357968932846404359363084, 5.50821445395345299948641768104, 6.79405906039010685182823940446, 7.71367077872044547263533331174, 8.758832935924856173060099057787, 9.196982768972178400591968116867, 10.02041446476298683486486191886

Graph of the ZZ-function along the critical line