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Properties

Label 2-525-105.104-c1-0-26
Degree 22
Conductor 525525
Sign −0.111+0.993i-0.111 + 0.993i
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  − 1.09·2-s + (0.323 − 1.70i)3-s − 0.791·4-s + (−0.355 + 1.87i)6-s + (2.44 + i)7-s + 3.06·8-s + (−2.79 − 1.09i)9-s − 3.06i·11-s + (−0.255 + 1.34i)12-s + 2.44·13-s + (−2.69 − 1.09i)14-s − 1.79·16-s − 2.69i·17-s + (3.06 + 1.20i)18-s + 4.38i·19-s + ⋯
L(s)  = 1  − 0.777·2-s + (0.186 − 0.982i)3-s − 0.395·4-s + (−0.144 + 0.763i)6-s + (0.925 + 0.377i)7-s + 1.08·8-s + (−0.930 − 0.366i)9-s − 0.925i·11-s + (−0.0737 + 0.388i)12-s + 0.679·13-s + (−0.719 − 0.293i)14-s − 0.447·16-s − 0.653i·17-s + (0.723 + 0.284i)18-s + 1.00i·19-s + ⋯

Functional equation

Λ(s)=(525s/2 ΓC(s) L(s)=((−0.111+0.993i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.111 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(525s/2 ΓC(s+1/2) L(s)=((−0.111+0.993i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.111 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 525525    =    3⋅52⋅73 \cdot 5^{2} \cdot 7
Sign: −0.111+0.993i-0.111 + 0.993i
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), −0.111+0.993i)(2,\ 525,\ (\ :1/2),\ -0.111 + 0.993i)

Particular Values

L(1)L(1) ≈\approx 0.616153−0.689029i0.616153 - 0.689029i
L(12)L(\frac12) ≈\approx 0.616153−0.689029i0.616153 - 0.689029i
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+(−0.323+1.70i)T 1 + (-0.323 + 1.70i)T
5 1 1
7 1+(−2.44−i)T 1 + (-2.44 - i)T
good2 1+1.09T+2T2 1 + 1.09T + 2T^{2}
11 1+3.06iT−11T2 1 + 3.06iT - 11T^{2}
13 1−2.44T+13T2 1 - 2.44T + 13T^{2}
17 1+2.69iT−17T2 1 + 2.69iT - 17T^{2}
19 1−4.38iT−19T2 1 - 4.38iT - 19T^{2}
23 1−5.26T+23T2 1 - 5.26T + 23T^{2}
29 1+5.26iT−29T2 1 + 5.26iT - 29T^{2}
31 1+6.83iT−31T2 1 + 6.83iT - 31T^{2}
37 1+8.58iT−37T2 1 + 8.58iT - 37T^{2}
41 1+10.2T+41T2 1 + 10.2T + 41T^{2}
43 1+6.58iT−43T2 1 + 6.58iT - 43T^{2}
47 1+2.69iT−47T2 1 + 2.69iT - 47T^{2}
53 1+3.93T+53T2 1 + 3.93T + 53T^{2}
59 1+7.51T+59T2 1 + 7.51T + 59T^{2}
61 1−6.83iT−61T2 1 - 6.83iT - 61T^{2}
67 1+4.16iT−67T2 1 + 4.16iT - 67T^{2}
71 1+3.06iT−71T2 1 + 3.06iT - 71T^{2}
73 1−16.1T+73T2 1 - 16.1T + 73T^{2}
79 1+0.582T+79T2 1 + 0.582T + 79T^{2}
83 1−15.5iT−83T2 1 - 15.5iT - 83T^{2}
89 1−7.51T+89T2 1 - 7.51T + 89T^{2}
97 1−11.7T+97T2 1 - 11.7T + 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

−10.72883620635928795584609752230, −9.410283173207150914955909502677, −8.674701637055155110648863285949, −8.093348190401457623250421818036, −7.38141699889761127649224929093, −6.07641155210349842967616788979, −5.16044681187479863854667454203, −3.66208082941647762765850933882, −2.04930744365160506444409857172, −0.790496798338438195516719997049, 1.50703478872330246170754156174, 3.38110662507074087999899644331, 4.72261686177246790873297109172, 4.94136395789514421132196476977, 6.76126748113426243016770535431, 7.88469856198650617324458870271, 8.630942104137068477181844126139, 9.230539670301388692039409617633, 10.24016913604915963668814315653, 10.74291524081581183741463249124

Graph of the ZZ-function along the critical line