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Properties

Label 2-2175-435.434-c0-0-10
Degree 22
Conductor 21752175
Sign −0.894+0.447i-0.894 + 0.447i
Analytic cond. 1.085461.08546
Root an. cond. 1.041851.04185
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s − i·3-s − 6-s + i·7-s − i·8-s − 9-s + 11-s − i·13-s + 14-s − 16-s − i·17-s + i·18-s + 21-s − i·22-s − 24-s + ⋯
L(s)  = 1  − i·2-s − i·3-s − 6-s + i·7-s − i·8-s − 9-s + 11-s − i·13-s + 14-s − 16-s − i·17-s + i·18-s + 21-s − i·22-s − 24-s + ⋯

Functional equation

Λ(s)=(2175s/2 ΓC(s) L(s)=((−0.894+0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2175s/2 ΓC(s) L(s)=((−0.894+0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21752175    =    3⋅52⋅293 \cdot 5^{2} \cdot 29
Sign: −0.894+0.447i-0.894 + 0.447i
Analytic conductor: 1.085461.08546
Root analytic conductor: 1.041851.04185
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2175(2174,⋅)\chi_{2175} (2174, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2175, ( :0), −0.894+0.447i)(2,\ 2175,\ (\ :0),\ -0.894 + 0.447i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.2995512781.299551278
L(12)L(\frac12) ≈\approx 1.2995512781.299551278
L(1)L(1) 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+iT 1 + iT
5 1 1
29 1−T 1 - T
good2 1+iT−T2 1 + iT - T^{2}
7 1−iT−T2 1 - iT - T^{2}
11 1−T+T2 1 - T + T^{2}
13 1+iT−T2 1 + iT - T^{2}
17 1+iT−T2 1 + iT - T^{2}
19 1−T2 1 - T^{2}
23 1+T2 1 + T^{2}
31 1−T2 1 - T^{2}
37 1+T2 1 + T^{2}
41 1+2T+T2 1 + 2T + T^{2}
43 1+T2 1 + T^{2}
47 1+iT−T2 1 + iT - T^{2}
53 1+T2 1 + T^{2}
59 1−T2 1 - T^{2}
61 1−T2 1 - T^{2}
67 1−iT−T2 1 - iT - T^{2}
71 1−T2 1 - T^{2}
73 1+T2 1 + T^{2}
79 1−T2 1 - T^{2}
83 1+T2 1 + T^{2}
89 1+T+T2 1 + T + T^{2}
97 1+T2 1 + 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

−8.907894277955556905030710725497, −8.332023881021735066899812635991, −7.25003020581366123991314221918, −6.65687536901623601143291557610, −5.86540235829576364993373479037, −4.96294781581387227309729325178, −3.52547111520363988912200447408, −2.81430228398164299149514404071, −2.03124391828388729288183065624, −0.947433202032195862022651083637, 1.73563806907604143759868119808, 3.22630213458761895300968613762, 4.17436563751000314271022318241, 4.70306895354367200128885015168, 5.80345566680843606687673426517, 6.53638663714794155429031410279, 7.04188134689925459582475490368, 8.120545201009249455083349146010, 8.664226170490811445719073850017, 9.472444300959686171008626663092

Graph of the ZZ-function along the critical line