↑

Properties

Label 1-39-39.38-r1-0-0
Degree 11
Conductor 3939
Sign 11
Analytic cond. 4.191134.19113
Root an. cond. 4.191134.19113
Motivic weight 00
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s + 11-s − 14-s + 16-s − 17-s − 19-s + 20-s + 22-s − 23-s + 25-s − 28-s − 29-s − 31-s + 32-s − 34-s − 35-s − 37-s − 38-s + 40-s + 41-s + 43-s + 44-s + ⋯
L(s)  = 1  + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s + 11-s − 14-s + 16-s − 17-s − 19-s + 20-s + 22-s − 23-s + 25-s − 28-s − 29-s − 31-s + 32-s − 34-s − 35-s − 37-s − 38-s + 40-s + 41-s + 43-s + 44-s + ⋯

Functional equation

Λ(s)=(39s/2 ΓR(s+1) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(39s/2 ΓR(s+1) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 3939    =    3⋅133 \cdot 13
Sign: 11
Analytic conductor: 4.191134.19113
Root analytic conductor: 4.191134.19113
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: χ39(38,⋅)\chi_{39} (38, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (1, 39, (1: ), 1)(1,\ 39,\ (1:\ ),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.7765814342.776581434
L(12)L(\frac12) ≈\approx 2.7765814342.776581434
L(1)L(1) ≈\approx 2.0122297262.012229726
L(1)L(1) ≈\approx 2.0122297262.012229726

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
13 1 1
good2 1+T 1 + T
5 1+T 1 + T
7 1−T 1 - T
11 1+T 1 + T
17 1−T 1 - T
19 1−T 1 - T
23 1−T 1 - T
29 1−T 1 - T
31 1−T 1 - T
37 1−T 1 - T
41 1+T 1 + T
43 1+T 1 + T
47 1+T 1 + T
53 1−T 1 - T
59 1+T 1 + T
61 1+T 1 + T
67 1−T 1 - T
71 1+T 1 + T
73 1−T 1 - T
79 1+T 1 + T
83 1+T 1 + T
89 1+T 1 + T
97 1−T 1 - T
show more
show less
   L(s)=∏p (1−αp p−s)−1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−34.7455728898393599410470802854, −33.36575152698929719192568831969, −32.654851397097391171163639940629, −31.63378022587838460339783881324, −30.071211749866349843068044079398, −29.38995764131271612067805504454, −28.21670238714452437561428321609, −26.080358633423214338465013260801, −25.2354638079280302768014451571, −24.092599756848773483522062032142, −22.47568677187093845556963849738, −21.93315191768140742388488531843, −20.48894258051107947462916786866, −19.28509033192190916774412727849, −17.337565702284410625359178605047, −16.14464333650454194819340125922, −14.63812553931791531619782220267, −13.467960588076771502489339540526, −12.46016779683215014834642518471, −10.75293335203503761289374520253, −9.26505664657555325046029148131, −6.79945605063326105935862116884, −5.85202180427453258469290681861, −3.955373319394198948596812494220, −2.15367543022529411299025556090, 2.15367543022529411299025556090, 3.955373319394198948596812494220, 5.85202180427453258469290681861, 6.79945605063326105935862116884, 9.26505664657555325046029148131, 10.75293335203503761289374520253, 12.46016779683215014834642518471, 13.467960588076771502489339540526, 14.63812553931791531619782220267, 16.14464333650454194819340125922, 17.337565702284410625359178605047, 19.28509033192190916774412727849, 20.48894258051107947462916786866, 21.93315191768140742388488531843, 22.47568677187093845556963849738, 24.092599756848773483522062032142, 25.2354638079280302768014451571, 26.080358633423214338465013260801, 28.21670238714452437561428321609, 29.38995764131271612067805504454, 30.071211749866349843068044079398, 31.63378022587838460339783881324, 32.654851397097391171163639940629, 33.36575152698929719192568831969, 34.7455728898393599410470802854

Graph of the ZZ-function along the critical line