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Properties

Label 2-525-5.4-c3-0-33
Degree 22
Conductor 525525
Sign 0.447−0.894i0.447 - 0.894i
Analytic cond. 30.976030.9760
Root an. cond. 5.565605.56560
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5i·2-s + 3i·3-s − 17·4-s − 15·6-s + 7i·7-s − 45i·8-s − 9·9-s + 12·11-s − 51i·12-s − 30i·13-s − 35·14-s + 89·16-s − 134i·17-s − 45i·18-s + 92·19-s + ⋯
L(s)  = 1  + 1.76i·2-s + 0.577i·3-s − 2.12·4-s − 1.02·6-s + 0.377i·7-s − 1.98i·8-s − 0.333·9-s + 0.328·11-s − 1.22i·12-s − 0.640i·13-s − 0.668·14-s + 1.39·16-s − 1.91i·17-s − 0.589i·18-s + 1.11·19-s + ⋯

Functional equation

Λ(s)=(525s/2 ΓC(s) L(s)=((0.447−0.894i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(525s/2 ΓC(s+3/2) L(s)=((0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 525525    =    3⋅52⋅73 \cdot 5^{2} \cdot 7
Sign: 0.447−0.894i0.447 - 0.894i
Analytic conductor: 30.976030.9760
Root analytic conductor: 5.565605.56560
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ525(274,⋅)\chi_{525} (274, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 525, ( :3/2), 0.447−0.894i)(2,\ 525,\ (\ :3/2),\ 0.447 - 0.894i)

Particular Values

L(2)L(2) ≈\approx 1.1667283811.166728381
L(12)L(\frac12) ≈\approx 1.1667283811.166728381
L(52)L(\frac{5}{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 1−3iT 1 - 3iT
5 1 1
7 1−7iT 1 - 7iT
good2 1−5iT−8T2 1 - 5iT - 8T^{2}
11 1−12T+1.33e3T2 1 - 12T + 1.33e3T^{2}
13 1+30iT−2.19e3T2 1 + 30iT - 2.19e3T^{2}
17 1+134iT−4.91e3T2 1 + 134iT - 4.91e3T^{2}
19 1−92T+6.85e3T2 1 - 92T + 6.85e3T^{2}
23 1+112iT−1.21e4T2 1 + 112iT - 1.21e4T^{2}
29 1−58T+2.43e4T2 1 - 58T + 2.43e4T^{2}
31 1+224T+2.97e4T2 1 + 224T + 2.97e4T^{2}
37 1+146iT−5.06e4T2 1 + 146iT - 5.06e4T^{2}
41 1−18T+6.89e4T2 1 - 18T + 6.89e4T^{2}
43 1+340iT−7.95e4T2 1 + 340iT - 7.95e4T^{2}
47 1−208iT−1.03e5T2 1 - 208iT - 1.03e5T^{2}
53 1−754iT−1.48e5T2 1 - 754iT - 1.48e5T^{2}
59 1+380T+2.05e5T2 1 + 380T + 2.05e5T^{2}
61 1−718T+2.26e5T2 1 - 718T + 2.26e5T^{2}
67 1−412iT−3.00e5T2 1 - 412iT - 3.00e5T^{2}
71 1+960T+3.57e5T2 1 + 960T + 3.57e5T^{2}
73 1+1.06e3iT−3.89e5T2 1 + 1.06e3iT - 3.89e5T^{2}
79 1+896T+4.93e5T2 1 + 896T + 4.93e5T^{2}
83 1+436iT−5.71e5T2 1 + 436iT - 5.71e5T^{2}
89 1−1.03e3T+7.04e5T2 1 - 1.03e3T + 7.04e5T^{2}
97 1+702iT−9.12e5T2 1 + 702iT - 9.12e5T^{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

−10.25933895455305968310546091819, −9.227145245004230714359924725357, −8.887350438553412334698872153535, −7.66615696543583308656028630782, −7.08702854628003906995801771606, −5.88921960399178111643206010912, −5.24910831615715771093284791001, −4.38375148537717703945141083609, −2.97792479975793590736842334955, −0.40926267412199916161492296753, 1.18480633670555135579211835896, 1.89776166478418392284370755303, 3.32461736289031074119610053593, 4.04907078036679444790608634026, 5.34578897225955341319423305355, 6.63258564425627448251898906408, 7.85908873739547548724140952246, 8.809394589780622704061818389405, 9.674178732454685000049996961408, 10.42595755866739323090356494508

Graph of the ZZ-function along the critical line