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Properties

Label 2-2175-87.86-c0-0-12
Degree 22
Conductor 21752175
Sign 0.707−0.707i0.707 - 0.707i
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·2-s + (0.707 + 0.707i)3-s + 1.00·4-s + (1.00 + 1.00i)6-s + 1.00i·9-s + (0.707 + 0.707i)12-s − 0.999·16-s + 1.41·17-s + 1.41i·18-s + (−0.707 + 0.707i)27-s − i·29-s − 1.41·32-s + 2.00·34-s + 1.00i·36-s + 1.41i·37-s + ⋯
L(s)  = 1  + 1.41·2-s + (0.707 + 0.707i)3-s + 1.00·4-s + (1.00 + 1.00i)6-s + 1.00i·9-s + (0.707 + 0.707i)12-s − 0.999·16-s + 1.41·17-s + 1.41i·18-s + (−0.707 + 0.707i)27-s − i·29-s − 1.41·32-s + 2.00·34-s + 1.00i·36-s + 1.41i·37-s + ⋯

Functional equation

Λ(s)=(2175s/2 ΓC(s) L(s)=((0.707−0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2175s/2 ΓC(s) L(s)=((0.707−0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21752175    =    3⋅52⋅293 \cdot 5^{2} \cdot 29
Sign: 0.707−0.707i0.707 - 0.707i
Analytic conductor: 1.085461.08546
Root analytic conductor: 1.041851.04185
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2175(1826,⋅)\chi_{2175} (1826, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2175, ( :0), 0.707−0.707i)(2,\ 2175,\ (\ :0),\ 0.707 - 0.707i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.9350009482.935000948
L(12)L(\frac12) ≈\approx 2.9350009482.935000948
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+(−0.707−0.707i)T 1 + (-0.707 - 0.707i)T
5 1 1
29 1+iT 1 + iT
good2 1−1.41T+T2 1 - 1.41T + T^{2}
7 1+T2 1 + T^{2}
11 1+T2 1 + T^{2}
13 1+T2 1 + T^{2}
17 1−1.41T+T2 1 - 1.41T + T^{2}
19 1−T2 1 - T^{2}
23 1−T2 1 - T^{2}
31 1−T2 1 - T^{2}
37 1−1.41iT−T2 1 - 1.41iT - T^{2}
41 1+T2 1 + T^{2}
43 1+1.41iT−T2 1 + 1.41iT - T^{2}
47 1+1.41T+T2 1 + 1.41T + T^{2}
53 1−T2 1 - T^{2}
59 1+2iT−T2 1 + 2iT - T^{2}
61 1−T2 1 - T^{2}
67 1+T2 1 + T^{2}
71 1+2iT−T2 1 + 2iT - T^{2}
73 1−1.41iT−T2 1 - 1.41iT - T^{2}
79 1−T2 1 - T^{2}
83 1−T2 1 - T^{2}
89 1+T2 1 + T^{2}
97 1+1.41iT−T2 1 + 1.41iT - 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

−9.556974442679638383688155829194, −8.463113385973294927159560564195, −7.87807054493720087823055555097, −6.82581002725355237866839870875, −5.90567529884247538508891719902, −5.12524807159713461958350129817, −4.53279872115235031684175581565, −3.54128174111645860228937893763, −3.12826316079474454733365428903, −1.97754050937281310951466859758, 1.47067445861492531573189272021, 2.72157005976622050768543625316, 3.34302047631195795110182512882, 4.15364416745678358955646277686, 5.19891902844400130386375685940, 5.91704177012419885978283356177, 6.69436151736711633518903983827, 7.45840512875108568941140390961, 8.210828407104074380499983712850, 9.109776717196761230132872383499

Graph of the ZZ-function along the critical line