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Properties

Label 1-61-61.60-r0-0-0
Degree 11
Conductor 6161
Sign 11
Analytic cond. 0.2832820.283282
Root an. cond. 0.2832820.283282
Motivic weight 00
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

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 − 11-s + 12-s + 13-s + 14-s + 15-s + 16-s − 17-s − 18-s + 19-s + 20-s − 21-s + 22-s − 23-s − 24-s + 25-s − 26-s + 27-s − 28-s + ⋯
L(s)  = 1  − 2-s + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s − 11-s + 12-s + 13-s + 14-s + 15-s + 16-s − 17-s − 18-s + 19-s + 20-s − 21-s + 22-s − 23-s − 24-s + 25-s − 26-s + 27-s − 28-s + ⋯

Functional equation

Λ(s)=(61s/2 ΓR(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(61s/2 ΓR(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 6161
Sign: 11
Analytic conductor: 0.2832820.283282
Root analytic conductor: 0.2832820.283282
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: χ61(60,⋅)\chi_{61} (60, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (1, 61, (0: ), 1)(1,\ 61,\ (0:\ ),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.84844025250.8484402525
L(12)L(\frac12) ≈\approx 0.84844025250.8484402525
L(1)L(1) ≈\approx 0.93831019820.9383101982
L(1)L(1) ≈\approx 0.93831019820.9383101982

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad61 1 1
good2 1−T 1 - T
3 1+T 1 + T
5 1+T 1 + T
7 1−T 1 - T
11 1−T 1 - T
13 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
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
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   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

−32.82573411853098146644267018675, −31.40500258932120590373403663653, −30.07959899261099683320680980451, −29.098555856325775051405658006562, −28.223749692000577024700664824476, −26.44569956073799372286794402195, −26.05503235862351310609443033328, −25.17795901381565984513361748684, −24.07370651255462611318217007938, −22.03419632198522342627092944428, −20.78392187781689933742302267641, −20.057441138757320313363140006188, −18.672598721958749196534732624781, −18.00576347603514123546512991036, −16.29315012627397787134601995636, −15.489382874162138740862518318787, −13.789131049820828862752138411306, −12.8133754336178107707525794242, −10.669447748894294158852750249409, −9.64611657141885916446348243181, −8.79080826384926706085041087801, −7.32744087186156568040361758365, −5.970300793360114507675288812948, −3.22773639167905402892302632100, −1.944591235003101921300981709370, 1.944591235003101921300981709370, 3.22773639167905402892302632100, 5.970300793360114507675288812948, 7.32744087186156568040361758365, 8.79080826384926706085041087801, 9.64611657141885916446348243181, 10.669447748894294158852750249409, 12.8133754336178107707525794242, 13.789131049820828862752138411306, 15.489382874162138740862518318787, 16.29315012627397787134601995636, 18.00576347603514123546512991036, 18.672598721958749196534732624781, 20.057441138757320313363140006188, 20.78392187781689933742302267641, 22.03419632198522342627092944428, 24.07370651255462611318217007938, 25.17795901381565984513361748684, 26.05503235862351310609443033328, 26.44569956073799372286794402195, 28.223749692000577024700664824476, 29.098555856325775051405658006562, 30.07959899261099683320680980451, 31.40500258932120590373403663653, 32.82573411853098146644267018675

Graph of the ZZ-function along the critical line