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Properties

Label 2-825-55.54-c2-0-61
Degree 22
Conductor 825825
Sign 0.161+0.986i0.161 + 0.986i
Analytic cond. 22.479622.4796
Root an. cond. 4.741264.74126
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  + 2.35·2-s + 1.73i·3-s + 1.53·4-s + 4.07i·6-s − 6.42·7-s − 5.79·8-s − 2.99·9-s + (3.26 − 10.5i)11-s + 2.66i·12-s + 7.68·13-s − 15.1·14-s − 19.7·16-s + 8.15·17-s − 7.05·18-s − 30.4i·19-s + ⋯
L(s)  = 1  + 1.17·2-s + 0.577i·3-s + 0.383·4-s + 0.679i·6-s − 0.918·7-s − 0.724·8-s − 0.333·9-s + (0.297 − 0.954i)11-s + 0.221i·12-s + 0.591·13-s − 1.08·14-s − 1.23·16-s + 0.479·17-s − 0.392·18-s − 1.60i·19-s + ⋯

Functional equation

Λ(s)=(825s/2 ΓC(s) L(s)=((0.161+0.986i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.161 + 0.986i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(825s/2 ΓC(s+1) L(s)=((0.161+0.986i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.161 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 825825    =    3⋅52⋅113 \cdot 5^{2} \cdot 11
Sign: 0.161+0.986i0.161 + 0.986i
Analytic conductor: 22.479622.4796
Root analytic conductor: 4.741264.74126
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ825(274,⋅)\chi_{825} (274, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 825, ( :1), 0.161+0.986i)(2,\ 825,\ (\ :1),\ 0.161 + 0.986i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.8903773651.890377365
L(12)L(\frac12) ≈\approx 1.8903773651.890377365
L(2)L(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 1−1.73iT 1 - 1.73iT
5 1 1
11 1+(−3.26+10.5i)T 1 + (-3.26 + 10.5i)T
good2 1−2.35T+4T2 1 - 2.35T + 4T^{2}
7 1+6.42T+49T2 1 + 6.42T + 49T^{2}
13 1−7.68T+169T2 1 - 7.68T + 169T^{2}
17 1−8.15T+289T2 1 - 8.15T + 289T^{2}
19 1+30.4iT−361T2 1 + 30.4iT - 361T^{2}
23 1+31.6iT−529T2 1 + 31.6iT - 529T^{2}
29 1−6.88iT−841T2 1 - 6.88iT - 841T^{2}
31 1−51.1T+961T2 1 - 51.1T + 961T^{2}
37 1−19.4iT−1.36e3T2 1 - 19.4iT - 1.36e3T^{2}
41 1+47.9iT−1.68e3T2 1 + 47.9iT - 1.68e3T^{2}
43 1+81.8T+1.84e3T2 1 + 81.8T + 1.84e3T^{2}
47 1+30.1iT−2.20e3T2 1 + 30.1iT - 2.20e3T^{2}
53 1−26.0iT−2.80e3T2 1 - 26.0iT - 2.80e3T^{2}
59 1+82.7T+3.48e3T2 1 + 82.7T + 3.48e3T^{2}
61 1+75.4iT−3.72e3T2 1 + 75.4iT - 3.72e3T^{2}
67 1−34iT−4.48e3T2 1 - 34iT - 4.48e3T^{2}
71 1+72.7T+5.04e3T2 1 + 72.7T + 5.04e3T^{2}
73 1−54.8T+5.32e3T2 1 - 54.8T + 5.32e3T^{2}
79 1−24.3iT−6.24e3T2 1 - 24.3iT - 6.24e3T^{2}
83 1−0.923T+6.88e3T2 1 - 0.923T + 6.88e3T^{2}
89 1+44.8T+7.92e3T2 1 + 44.8T + 7.92e3T^{2}
97 1−21.2iT−9.40e3T2 1 - 21.2iT - 9.40e3T^{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

−9.879961805461761980513997874137, −8.964625365235626545709181559195, −8.405957290960562350926540130524, −6.66862589048834094300584837630, −6.27735447699628078763967396242, −5.21499507098616068580958392760, −4.40833735398219060646882626141, −3.38638694402186271955960974073, −2.83373507663556692658547598292, −0.42648413184385779367089645420, 1.54321745607103305930250826589, 3.03765686614352284075997226973, 3.75332230209041089257256637853, 4.83339748008696943710181872572, 5.97683487938385396093107649922, 6.35890724284001835498404511365, 7.44701100534141527123991929900, 8.397762765531320726812767216366, 9.560438199114060771204030959959, 10.06025520142869752925996857542

Graph of the ZZ-function along the critical line