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Properties

Label 2-33e2-33.32-c1-0-0
Degree 22
Conductor 10891089
Sign −0.870−0.492i-0.870 - 0.492i
Analytic cond. 8.695708.69570
Root an. cond. 2.948842.94884
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s + 1.41i·5-s − 2.82i·7-s − 3·8-s + 1.41i·10-s + 4.24i·13-s − 2.82i·14-s − 16-s − 6·17-s − 1.41i·20-s + 8.48i·23-s + 2.99·25-s + 4.24i·26-s + 2.82i·28-s − 8·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.5·4-s + 0.632i·5-s − 1.06i·7-s − 1.06·8-s + 0.447i·10-s + 1.17i·13-s − 0.755i·14-s − 0.250·16-s − 1.45·17-s − 0.316i·20-s + 1.76i·23-s + 0.599·25-s + 0.832i·26-s + 0.534i·28-s − 1.48·29-s + ⋯

Functional equation

Λ(s)=(1089s/2 ΓC(s) L(s)=((−0.870−0.492i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.870 - 0.492i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1089s/2 ΓC(s+1/2) L(s)=((−0.870−0.492i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.870 - 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10891089    =    32⋅1123^{2} \cdot 11^{2}
Sign: −0.870−0.492i-0.870 - 0.492i
Analytic conductor: 8.695708.69570
Root analytic conductor: 2.948842.94884
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1089(1088,⋅)\chi_{1089} (1088, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1089, ( :1/2), −0.870−0.492i)(2,\ 1089,\ (\ :1/2),\ -0.870 - 0.492i)

Particular Values

L(1)L(1) ≈\approx 0.56553343620.5655334362
L(12)L(\frac12) ≈\approx 0.56553343620.5655334362
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)
bad3 1 1
11 1 1
good2 1−T+2T2 1 - T + 2T^{2}
5 1−1.41iT−5T2 1 - 1.41iT - 5T^{2}
7 1+2.82iT−7T2 1 + 2.82iT - 7T^{2}
13 1−4.24iT−13T2 1 - 4.24iT - 13T^{2}
17 1+6T+17T2 1 + 6T + 17T^{2}
19 1−19T2 1 - 19T^{2}
23 1−8.48iT−23T2 1 - 8.48iT - 23T^{2}
29 1+8T+29T2 1 + 8T + 29T^{2}
31 1+8T+31T2 1 + 8T + 31T^{2}
37 1+10T+37T2 1 + 10T + 37T^{2}
41 1+41T2 1 + 41T^{2}
43 1−43T2 1 - 43T^{2}
47 1+2.82iT−47T2 1 + 2.82iT - 47T^{2}
53 1+1.41iT−53T2 1 + 1.41iT - 53T^{2}
59 1−11.3iT−59T2 1 - 11.3iT - 59T^{2}
61 1+7.07iT−61T2 1 + 7.07iT - 61T^{2}
67 1+4T+67T2 1 + 4T + 67T^{2}
71 1−14.1iT−71T2 1 - 14.1iT - 71T^{2}
73 1+7.07iT−73T2 1 + 7.07iT - 73T^{2}
79 1+8.48iT−79T2 1 + 8.48iT - 79T^{2}
83 1+4T+83T2 1 + 4T + 83T^{2}
89 1+4.24iT−89T2 1 + 4.24iT - 89T^{2}
97 1−10T+97T2 1 - 10T + 97T^{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

−10.28386302674145506190150135339, −9.270767444191115457050762504746, −8.821530432246395056577364210723, −7.33527517311975019436416413494, −6.97378168104924839469547757038, −5.89082107297916174387086381749, −4.89830335170751826937406022994, −3.97494312672525015030931134646, −3.44197402475290713933458150761, −1.87541829871856476581948085132, 0.19026246820860583551012464840, 2.20554667488558793374068410950, 3.30344458171355219951648282808, 4.41770025287563331225164272071, 5.20396691560226240479717611412, 5.76871584237633525626608958480, 6.78313108307094394755798447436, 8.121550675525604134772370416068, 8.937983384080349493656413036513, 9.087382126779537228013400247292

Graph of the ZZ-function along the critical line