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Properties

Label 2-525-105.104-c1-0-21
Degree 22
Conductor 525525
Sign −0.129+0.991i-0.129 + 0.991i
Analytic cond. 4.192144.19214
Root an. cond. 2.047472.04747
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  − 1.73·3-s − 2·4-s + (0.866 + 2.5i)7-s + 2.99·9-s + 3.46·12-s − 5.19·13-s + 4·16-s − 8.66i·19-s + (−1.49 − 4.33i)21-s − 5.19·27-s + (−1.73 − 5i)28-s − 8.66i·31-s − 5.99·36-s − 10i·37-s + 9·39-s + ⋯
L(s)  = 1  − 1.00·3-s − 4-s + (0.327 + 0.944i)7-s + 0.999·9-s + 1.00·12-s − 1.44·13-s + 16-s − 1.98i·19-s + (−0.327 − 0.944i)21-s − 1.00·27-s + (−0.327 − 0.944i)28-s − 1.55i·31-s − 0.999·36-s − 1.64i·37-s + 1.44·39-s + ⋯

Functional equation

Λ(s)=(525s/2 ΓC(s) L(s)=((−0.129+0.991i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.129 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(525s/2 ΓC(s+1/2) L(s)=((−0.129+0.991i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.129 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 525525    =    3⋅52⋅73 \cdot 5^{2} \cdot 7
Sign: −0.129+0.991i-0.129 + 0.991i
Analytic conductor: 4.192144.19214
Root analytic conductor: 2.047472.04747
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ525(524,⋅)\chi_{525} (524, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 525, ( :1/2), −0.129+0.991i)(2,\ 525,\ (\ :1/2),\ -0.129 + 0.991i)

Particular Values

L(1)L(1) ≈\approx 0.289103−0.329418i0.289103 - 0.329418i
L(12)L(\frac12) ≈\approx 0.289103−0.329418i0.289103 - 0.329418i
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+1.73T 1 + 1.73T
5 1 1
7 1+(−0.866−2.5i)T 1 + (-0.866 - 2.5i)T
good2 1+2T2 1 + 2T^{2}
11 1−11T2 1 - 11T^{2}
13 1+5.19T+13T2 1 + 5.19T + 13T^{2}
17 1−17T2 1 - 17T^{2}
19 1+8.66iT−19T2 1 + 8.66iT - 19T^{2}
23 1+23T2 1 + 23T^{2}
29 1−29T2 1 - 29T^{2}
31 1+8.66iT−31T2 1 + 8.66iT - 31T^{2}
37 1+10iT−37T2 1 + 10iT - 37T^{2}
41 1+41T2 1 + 41T^{2}
43 1+5iT−43T2 1 + 5iT - 43T^{2}
47 1−47T2 1 - 47T^{2}
53 1+53T2 1 + 53T^{2}
59 1+59T2 1 + 59T^{2}
61 1−8.66iT−61T2 1 - 8.66iT - 61T^{2}
67 1+5iT−67T2 1 + 5iT - 67T^{2}
71 1−71T2 1 - 71T^{2}
73 1−13.8T+73T2 1 - 13.8T + 73T^{2}
79 1−4T+79T2 1 - 4T + 79T^{2}
83 1−83T2 1 - 83T^{2}
89 1+89T2 1 + 89T^{2}
97 1+19.0T+97T2 1 + 19.0T + 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

−10.69065134712188122127316533885, −9.542994875702193078198259682580, −9.170377392242587473362865031441, −7.892373657994132158383364893483, −6.94528768279055183720331696645, −5.65537697477364257038411022252, −5.05521707132237474578327744846, −4.23950650292507199390242448515, −2.40628662692044119453696030807, −0.33090966372630186139671030315, 1.32405775916972774205726664311, 3.63816324703329082798230012468, 4.65708942210780292188596659207, 5.25319537156480419360196590242, 6.48180853456505477535957617944, 7.52691468897359460854277318247, 8.273361313864786450872978086708, 9.815122855065261890362439773622, 10.02746380944785580204169477457, 10.97597614297880465032724013415

Graph of the ZZ-function along the critical line