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Properties

Label 2-3549-273.263-c0-0-2
Degree 22
Conductor 35493549
Sign 0.990−0.139i0.990 - 0.139i
Analytic cond. 1.771181.77118
Root an. cond. 1.330851.33085
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + (−0.5 − 0.866i)4-s + (0.866 + 0.5i)7-s + 9-s + (0.5 + 0.866i)12-s + (−0.499 + 0.866i)16-s + 1.73·19-s + (−0.866 − 0.5i)21-s + (−0.5 + 0.866i)25-s − 27-s − 0.999i·28-s + (−0.5 − 0.866i)36-s + (−0.866 + 1.5i)37-s + (−0.5 + 0.866i)43-s + (0.499 − 0.866i)48-s + (0.499 + 0.866i)49-s + ⋯
L(s)  = 1  − 3-s + (−0.5 − 0.866i)4-s + (0.866 + 0.5i)7-s + 9-s + (0.5 + 0.866i)12-s + (−0.499 + 0.866i)16-s + 1.73·19-s + (−0.866 − 0.5i)21-s + (−0.5 + 0.866i)25-s − 27-s − 0.999i·28-s + (−0.5 − 0.866i)36-s + (−0.866 + 1.5i)37-s + (−0.5 + 0.866i)43-s + (0.499 − 0.866i)48-s + (0.499 + 0.866i)49-s + ⋯

Functional equation

Λ(s)=(3549s/2 ΓC(s) L(s)=((0.990−0.139i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3549s/2 ΓC(s) L(s)=((0.990−0.139i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 35493549    =    3⋅7⋅1323 \cdot 7 \cdot 13^{2}
Sign: 0.990−0.139i0.990 - 0.139i
Analytic conductor: 1.771181.77118
Root analytic conductor: 1.330851.33085
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3549(2174,⋅)\chi_{3549} (2174, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3549, ( :0), 0.990−0.139i)(2,\ 3549,\ (\ :0),\ 0.990 - 0.139i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.89217941060.8921794106
L(12)L(\frac12) ≈\approx 0.89217941060.8921794106
L(1)L(1) 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+T 1 + T
7 1+(−0.866−0.5i)T 1 + (-0.866 - 0.5i)T
13 1 1
good2 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
5 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
11 1−T2 1 - T^{2}
17 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
19 1−1.73T+T2 1 - 1.73T + T^{2}
23 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
29 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
31 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
37 1+(0.866−1.5i)T+(−0.5−0.866i)T2 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2}
41 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
43 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
47 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
53 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
59 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
61 1−T+T2 1 - T + T^{2}
67 1+T2 1 + T^{2}
71 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
73 1+(−0.866+1.5i)T+(−0.5−0.866i)T2 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2}
79 1+(−1−1.73i)T+(−0.5+0.866i)T2 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2}
83 1−T2 1 - T^{2}
89 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
97 1+(−0.866+1.5i)T+(−0.5−0.866i)T2 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{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

−8.854493954420810501310521651293, −7.997266133602546652588713713350, −7.19663746893793478089787504008, −6.34038179679990419652841850851, −5.54398975601875905315996244358, −5.14130230764624880779601567602, −4.53342165615795607515502366629, −3.39115628028124324791652960012, −1.84481440351503140923931631100, −1.08340420957720380407399075401, 0.77440089446680880906395584582, 2.10970033671160731325855042671, 3.53516288693393150077187806848, 4.12440998102278788896947307124, 5.04748239900065121897531305448, 5.43386561857907410451119919919, 6.61548764226214595588877172002, 7.40314273065301028948618159004, 7.76307429010896620177476744628, 8.659853058124347021694441080486

Graph of the ZZ-function along the critical line