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Properties

Label 2-1638-1.1-c1-0-23
Degree 22
Conductor 16381638
Sign −1-1
Analytic cond. 13.079413.0794
Root an. cond. 3.616553.61655
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 + 7-s − 8-s + 13-s − 14-s + 16-s − 6·17-s − 4·19-s − 6·23-s − 5·25-s − 26-s + 28-s + 8·31-s − 32-s + 6·34-s + 2·37-s + 4·38-s − 4·43-s + 6·46-s − 6·47-s + 49-s + 5·50-s + 52-s − 56-s − 6·59-s + 2·61-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.917·19-s − 1.25·23-s − 25-s − 0.196·26-s + 0.188·28-s + 1.43·31-s − 0.176·32-s + 1.02·34-s + 0.328·37-s + 0.648·38-s − 0.609·43-s + 0.884·46-s − 0.875·47-s + 1/7·49-s + 0.707·50-s + 0.138·52-s − 0.133·56-s − 0.781·59-s + 0.256·61-s + ⋯

Functional equation

Λ(s)=(1638s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1638s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16381638    =    2⋅32⋅7⋅132 \cdot 3^{2} \cdot 7 \cdot 13
Sign: −1-1
Analytic conductor: 13.079413.0794
Root analytic conductor: 3.616553.61655
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1638, ( :1/2), −1)(2,\ 1638,\ (\ :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 1
7 1−T 1 - T
13 1−T 1 - T
good5 1+pT2 1 + p T^{2}
11 1+pT2 1 + 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+6T+pT2 1 + 6 T + p T^{2}
29 1+pT2 1 + 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+4T+pT2 1 + 4 T + p T^{2}
47 1+6T+pT2 1 + 6 T + p T^{2}
53 1+pT2 1 + 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+4T+pT2 1 + 4 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+10T+pT2 1 + 10 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1+6T+pT2 1 + 6 T + p T^{2}
89 1+12T+pT2 1 + 12 T + p T^{2}
97 1−2T+pT2 1 - 2 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

−8.833123066969410272291086657384, −8.315380262939132711743531410580, −7.59113649640036618050579447442, −6.52107659919036530841891307507, −6.05514828155563894984871537986, −4.73802182050113221543979801436, −3.94161510269754508782544806387, −2.55291695292451487866541530230, −1.66895190682184687500782568597, 0, 1.66895190682184687500782568597, 2.55291695292451487866541530230, 3.94161510269754508782544806387, 4.73802182050113221543979801436, 6.05514828155563894984871537986, 6.52107659919036530841891307507, 7.59113649640036618050579447442, 8.315380262939132711743531410580, 8.833123066969410272291086657384

Graph of the ZZ-function along the critical line