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Properties

Label 2-2175-435.434-c0-0-15
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  − 1.87i·2-s − i·3-s − 2.53·4-s − 1.87·6-s − 1.53i·7-s + 2.87i·8-s − 9-s − 0.347·11-s + 2.53i·12-s − 1.87i·13-s − 2.87·14-s + 2.87·16-s + 0.347i·17-s + 1.87i·18-s − 1.53·21-s + 0.652i·22-s + ⋯
L(s)  = 1  − 1.87i·2-s − i·3-s − 2.53·4-s − 1.87·6-s − 1.53i·7-s + 2.87i·8-s − 9-s − 0.347·11-s + 2.53i·12-s − 1.87i·13-s − 2.87·14-s + 2.87·16-s + 0.347i·17-s + 1.87i·18-s − 1.53·21-s + 0.652i·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 0.76991131440.7699113144
L(12)L(\frac12) ≈\approx 0.76991131440.7699113144
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+1.87iT−T2 1 + 1.87iT - T^{2}
7 1+1.53iT−T2 1 + 1.53iT - T^{2}
11 1+0.347T+T2 1 + 0.347T + T^{2}
13 1+1.87iT−T2 1 + 1.87iT - T^{2}
17 1−0.347iT−T2 1 - 0.347iT - 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 1−1.53iT−T2 1 - 1.53iT - T^{2}
53 1+T2 1 + T^{2}
59 1−T2 1 - T^{2}
61 1−T2 1 - T^{2}
67 1+0.347iT−T2 1 + 0.347iT - 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+1.87T+T2 1 + 1.87T + 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

−8.647937042684188351803014590194, −7.961737743587762001713005965490, −7.42376918599719405307017080864, −6.13378364356315335638082733306, −5.15863694830878826220903695214, −4.22692813553507140304874873045, −3.25624927978870964100807954002, −2.66794085219172610065449103803, −1.35548121006515235899879292803, −0.59307221157721873904594030647, 2.46164880092221341098034529901, 3.82006731852525174849727657097, 4.69381539196767019473620969287, 5.20672566668728024673586175719, 6.02033580658454346769357605457, 6.56914893171827255655266655194, 7.54605390217298435507639834430, 8.631133837735891416794324199487, 8.787358693339412918670150294845, 9.461594426662689311063538568132

Graph of the ZZ-function along the critical line