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Properties

Label 2-2175-5.4-c1-0-64
Degree 22
Conductor 21752175
Sign −0.894+0.447i-0.894 + 0.447i
Analytic cond. 17.367417.3674
Root an. cond. 4.167424.16742
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.79i·2-s + i·3-s − 1.20·4-s + 1.79·6-s − i·7-s − 1.41i·8-s − 9-s + 5·11-s − 1.20i·12-s − 4.58i·13-s − 1.79·14-s − 4.95·16-s + 3i·17-s + 1.79i·18-s − 3.58·19-s + ⋯
L(s)  = 1  − 1.26i·2-s + 0.577i·3-s − 0.604·4-s + 0.731·6-s − 0.377i·7-s − 0.501i·8-s − 0.333·9-s + 1.50·11-s − 0.348i·12-s − 1.27i·13-s − 0.478·14-s − 1.23·16-s + 0.727i·17-s + 0.422i·18-s − 0.821·19-s + ⋯

Functional equation

Λ(s)=(2175s/2 ΓC(s) L(s)=((−0.894+0.447i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2175s/2 ΓC(s+1/2) L(s)=((−0.894+0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, 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: 17.367417.3674
Root analytic conductor: 4.167424.16742
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2175(349,⋅)\chi_{2175} (349, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2175, ( :1/2), −0.894+0.447i)(2,\ 2175,\ (\ :1/2),\ -0.894 + 0.447i)

Particular Values

L(1)L(1) ≈\approx 1.6581038811.658103881
L(12)L(\frac12) ≈\approx 1.6581038811.658103881
L(32)L(\frac{3}{2}) 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+1.79iT−2T2 1 + 1.79iT - 2T^{2}
7 1+iT−7T2 1 + iT - 7T^{2}
11 1−5T+11T2 1 - 5T + 11T^{2}
13 1+4.58iT−13T2 1 + 4.58iT - 13T^{2}
17 1−3iT−17T2 1 - 3iT - 17T^{2}
19 1+3.58T+19T2 1 + 3.58T + 19T^{2}
23 1+4iT−23T2 1 + 4iT - 23T^{2}
31 1−4T+31T2 1 - 4T + 31T^{2}
37 1−4iT−37T2 1 - 4iT - 37T^{2}
41 1+9.16T+41T2 1 + 9.16T + 41T^{2}
43 1+9.58iT−43T2 1 + 9.58iT - 43T^{2}
47 1+10.5iT−47T2 1 + 10.5iT - 47T^{2}
53 1−0.417iT−53T2 1 - 0.417iT - 53T^{2}
59 1−7.58T+59T2 1 - 7.58T + 59T^{2}
61 1−12.7T+61T2 1 - 12.7T + 61T^{2}
67 1−4.16iT−67T2 1 - 4.16iT - 67T^{2}
71 1+9.58T+71T2 1 + 9.58T + 71T^{2}
73 1−4iT−73T2 1 - 4iT - 73T^{2}
79 1+7.58T+79T2 1 + 7.58T + 79T^{2}
83 1+11.5iT−83T2 1 + 11.5iT - 83T^{2}
89 1+1.41T+89T2 1 + 1.41T + 89T^{2}
97 1+11.5iT−97T2 1 + 11.5iT - 97T^{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

−8.753101601478421218480050782184, −8.476225949962167607138452332631, −7.04388804678459295085891399283, −6.42693224397819184236512680534, −5.35506857290199480795398613436, −4.16357382058152508737311638387, −3.82146709273995111677361717207, −2.87039569639614589938371165287, −1.76332515639060866721098504392, −0.58698656628805986390457317485, 1.46564327400613206822686900070, 2.49355935340586187300617968016, 3.92994855302328972088214128811, 4.79646561414338014838608766048, 5.78772599731370243859779649171, 6.52025793371046685441787730877, 6.85675038928827682182166198132, 7.65522005202914575443512844989, 8.552990298507929512903568142319, 9.065106364759053554227844637744

Graph of the ZZ-function along the critical line