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Properties

Label 2-2175-435.434-c0-0-13
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  − 0.347i·2-s + i·3-s + 0.879·4-s + 0.347·6-s − 1.87i·7-s − 0.652i·8-s − 9-s − 1.53·11-s + 0.879i·12-s − 0.347i·13-s − 0.652·14-s + 0.652·16-s − 1.53i·17-s + 0.347i·18-s + 1.87·21-s + 0.532i·22-s + ⋯
L(s)  = 1  − 0.347i·2-s + i·3-s + 0.879·4-s + 0.347·6-s − 1.87i·7-s − 0.652i·8-s − 9-s − 1.53·11-s + 0.879i·12-s − 0.347i·13-s − 0.652·14-s + 0.652·16-s − 1.53i·17-s + 0.347i·18-s + 1.87·21-s + 0.532i·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 1.2325225141.232522514
L(12)L(\frac12) ≈\approx 1.2325225141.232522514
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+0.347iT−T2 1 + 0.347iT - T^{2}
7 1+1.87iT−T2 1 + 1.87iT - T^{2}
11 1+1.53T+T2 1 + 1.53T + T^{2}
13 1+0.347iT−T2 1 + 0.347iT - T^{2}
17 1+1.53iT−T2 1 + 1.53iT - 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 1−1.87iT−T2 1 - 1.87iT - T^{2}
53 1+T2 1 + T^{2}
59 1−T2 1 - T^{2}
61 1−T2 1 - T^{2}
67 1−1.53iT−T2 1 - 1.53iT - 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 1−0.347T+T2 1 - 0.347T + 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

−9.554441044523615279202614159107, −8.174871064809424412150866223821, −7.56871105815980012134461545197, −6.95576065887986242762956142598, −5.88289268434447235264471435553, −4.91249079498786374679198883511, −4.23406977267204944449113408011, −3.16310656184559475310269913081, −2.63978715530605962633459164257, −0.811823675829709997785431308179, 1.87237842216213082542263440373, 2.38481037981051107665735447477, 3.17888835556466171006101325012, 5.04957346393575503913048518832, 5.74647972366533448638752400979, 6.18074493988751036527167172600, 6.99113733584762074763094910187, 8.063214235273163556096168692161, 8.224500300559437847655053487270, 9.047179885302402289033225240344

Graph of the ZZ-function along the critical line