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Properties

Label 2-2175-435.434-c0-0-6
Degree 22
Conductor 21752175
Sign 0.447−0.894i0.447 - 0.894i
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  + 1.53i·2-s − i·3-s − 1.34·4-s + 1.53·6-s − 0.347i·7-s − 0.532i·8-s − 9-s + 1.87·11-s + 1.34i·12-s + 1.53i·13-s + 0.532·14-s − 0.532·16-s − 1.87i·17-s − 1.53i·18-s − 0.347·21-s + 2.87i·22-s + ⋯
L(s)  = 1  + 1.53i·2-s − i·3-s − 1.34·4-s + 1.53·6-s − 0.347i·7-s − 0.532i·8-s − 9-s + 1.87·11-s + 1.34i·12-s + 1.53i·13-s + 0.532·14-s − 0.532·16-s − 1.87i·17-s − 1.53i·18-s − 0.347·21-s + 2.87i·22-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 21752175    =    3⋅52⋅293 \cdot 5^{2} \cdot 29
Sign: 0.447−0.894i0.447 - 0.894i
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), 0.447−0.894i)(2,\ 2175,\ (\ :0),\ 0.447 - 0.894i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.2058810561.205881056
L(12)L(\frac12) ≈\approx 1.2058810561.205881056
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 1−1.53iT−T2 1 - 1.53iT - T^{2}
7 1+0.347iT−T2 1 + 0.347iT - T^{2}
11 1−1.87T+T2 1 - 1.87T + T^{2}
13 1−1.53iT−T2 1 - 1.53iT - T^{2}
17 1+1.87iT−T2 1 + 1.87iT - 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−T+T2 1 - T + T^{2}
43 1+T2 1 + T^{2}
47 1−0.347iT−T2 1 - 0.347iT - T^{2}
53 1+T2 1 + T^{2}
59 1−T2 1 - T^{2}
61 1−T2 1 - T^{2}
67 1−1.87iT−T2 1 - 1.87iT - 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 1−1.53T+T2 1 - 1.53T + 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

−9.120114399174222850121149402071, −8.493859891323851030355008546614, −7.46041159003987757992703552497, −6.94751150768018472852181300659, −6.60854051057448722831774981161, −5.83848832632293935225825974135, −4.77184252945360427516941172874, −4.01880319095264500850401484173, −2.51474962277231355101912560480, −1.16097646635749246192058199182, 1.14951364616999161823297943983, 2.39482021786442359122130640401, 3.44912402999377911778468326972, 3.85693782182176314157368440837, 4.70964438917740111320191090345, 5.81673649232392043734986820544, 6.48835568156517295865761999221, 8.038717719081150312969326345698, 8.761720593175655671886089939897, 9.315863832198827647008157689955

Graph of the ZZ-function along the critical line