Properties

Label 2-525-15.14-c2-0-10
Degree 22
Conductor 525525
Sign 0.7370.675i-0.737 - 0.675i
Analytic cond. 14.305214.3052
Root an. cond. 3.782223.78222
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.50·2-s + (−2.88 − 0.822i)3-s + 8.29·4-s + (10.1 + 2.88i)6-s + 2.64i·7-s − 15.0·8-s + (7.64 + 4.74i)9-s + 7.01i·11-s + (−23.9 − 6.82i)12-s + 11.6i·13-s − 9.27i·14-s + 19.5·16-s + 4.52·17-s + (−26.8 − 16.6i)18-s − 16.2·19-s + ⋯
L(s)  = 1  − 1.75·2-s + (−0.961 − 0.274i)3-s + 2.07·4-s + (1.68 + 0.480i)6-s + 0.377i·7-s − 1.88·8-s + (0.849 + 0.527i)9-s + 0.637i·11-s + (−1.99 − 0.568i)12-s + 0.895i·13-s − 0.662i·14-s + 1.22·16-s + 0.266·17-s + (−1.48 − 0.924i)18-s − 0.854·19-s + ⋯

Functional equation

Λ(s)=(525s/2ΓC(s)L(s)=((0.7370.675i)Λ(3s)\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.737 - 0.675i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(525s/2ΓC(s+1)L(s)=((0.7370.675i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.737 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 525525    =    35273 \cdot 5^{2} \cdot 7
Sign: 0.7370.675i-0.737 - 0.675i
Analytic conductor: 14.305214.3052
Root analytic conductor: 3.782223.78222
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ525(449,)\chi_{525} (449, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 525, ( :1), 0.7370.675i)(2,\ 525,\ (\ :1),\ -0.737 - 0.675i)

Particular Values

L(32)L(\frac{3}{2}) \approx 0.25090217940.2509021794
L(12)L(\frac12) \approx 0.25090217940.2509021794
L(2)L(2) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1+(2.88+0.822i)T 1 + (2.88 + 0.822i)T
5 1 1
7 12.64iT 1 - 2.64iT
good2 1+3.50T+4T2 1 + 3.50T + 4T^{2}
11 17.01iT121T2 1 - 7.01iT - 121T^{2}
13 111.6iT169T2 1 - 11.6iT - 169T^{2}
17 14.52T+289T2 1 - 4.52T + 289T^{2}
19 1+16.2T+361T2 1 + 16.2T + 361T^{2}
23 125.5T+529T2 1 - 25.5T + 529T^{2}
29 19.49iT841T2 1 - 9.49iT - 841T^{2}
31 128.7T+961T2 1 - 28.7T + 961T^{2}
37 1+33.0iT1.36e3T2 1 + 33.0iT - 1.36e3T^{2}
41 1+67.1iT1.68e3T2 1 + 67.1iT - 1.68e3T^{2}
43 124.1iT1.84e3T2 1 - 24.1iT - 1.84e3T^{2}
47 1+33.0T+2.20e3T2 1 + 33.0T + 2.20e3T^{2}
53 115.1T+2.80e3T2 1 - 15.1T + 2.80e3T^{2}
59 192.3iT3.48e3T2 1 - 92.3iT - 3.48e3T^{2}
61 1+57.5T+3.72e3T2 1 + 57.5T + 3.72e3T^{2}
67 115.1iT4.48e3T2 1 - 15.1iT - 4.48e3T^{2}
71 170.5iT5.04e3T2 1 - 70.5iT - 5.04e3T^{2}
73 176.7iT5.32e3T2 1 - 76.7iT - 5.32e3T^{2}
79 1+127.T+6.24e3T2 1 + 127.T + 6.24e3T^{2}
83 1+74.2T+6.88e3T2 1 + 74.2T + 6.88e3T^{2}
89 1+127.iT7.92e3T2 1 + 127. iT - 7.92e3T^{2}
97 1+23.1iT9.40e3T2 1 + 23.1iT - 9.40e3T^{2}
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   L(s)=p j=12(1αj,pps)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

−10.79248929326776951025585014227, −10.11609451384325214113471507073, −9.233564641713618833138052718106, −8.472454276649004073935737831114, −7.30181138622171752916101680263, −6.84359794821081223035319738105, −5.81472945787175058487773011994, −4.47797966368383683688004744060, −2.35594460415009739122921186981, −1.26644693876941901087365725537, 0.23989718834349543539577315248, 1.28365855846023892076999212243, 3.09363940805499963968038577497, 4.75569676792971331114875772475, 6.07325040311523508925414051615, 6.76933123819680012363827147818, 7.81290359052971347423357053952, 8.554603133307914506661868828484, 9.639303347457660926720929161704, 10.24653333164115489065411647573

Graph of the ZZ-function along the critical line