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Properties

Label 2-1458-3.2-c2-0-0
Degree 22
Conductor 14581458
Sign −1-1
Analytic cond. 39.727639.7276
Root an. cond. 6.302986.30298
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  − 1.41i·2-s − 2.00·4-s + 7.93i·5-s + 0.453·7-s + 2.82i·8-s + 11.2·10-s − 14.1i·11-s − 12.8·13-s − 0.641i·14-s + 4.00·16-s + 32.9i·17-s − 0.405·19-s − 15.8i·20-s − 19.9·22-s − 14.4i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.500·4-s + 1.58i·5-s + 0.0648·7-s + 0.353i·8-s + 1.12·10-s − 1.28i·11-s − 0.984·13-s − 0.0458i·14-s + 0.250·16-s + 1.93i·17-s − 0.0213·19-s − 0.793i·20-s − 0.906·22-s − 0.629i·23-s + ⋯

Functional equation

Λ(s)=(1458s/2 ΓC(s) L(s)=(− Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(1458s/2 ΓC(s+1) L(s)=(− Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14581458    =    2⋅362 \cdot 3^{6}
Sign: −1-1
Analytic conductor: 39.727639.7276
Root analytic conductor: 6.302986.30298
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ1458(1457,⋅)\chi_{1458} (1457, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1458, ( :1), −1)(2,\ 1458,\ (\ :1),\ -1)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.0061833352340.006183335234
L(12)L(\frac12) ≈\approx 0.0061833352340.006183335234
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)
bad2 1+1.41iT 1 + 1.41iT
3 1 1
good5 1−7.93iT−25T2 1 - 7.93iT - 25T^{2}
7 1−0.453T+49T2 1 - 0.453T + 49T^{2}
11 1+14.1iT−121T2 1 + 14.1iT - 121T^{2}
13 1+12.8T+169T2 1 + 12.8T + 169T^{2}
17 1−32.9iT−289T2 1 - 32.9iT - 289T^{2}
19 1+0.405T+361T2 1 + 0.405T + 361T^{2}
23 1+14.4iT−529T2 1 + 14.4iT - 529T^{2}
29 1−26.2iT−841T2 1 - 26.2iT - 841T^{2}
31 1−24.9T+961T2 1 - 24.9T + 961T^{2}
37 1+7.68T+1.36e3T2 1 + 7.68T + 1.36e3T^{2}
41 1+24.7iT−1.68e3T2 1 + 24.7iT - 1.68e3T^{2}
43 1−17.9T+1.84e3T2 1 - 17.9T + 1.84e3T^{2}
47 1+47.4iT−2.20e3T2 1 + 47.4iT - 2.20e3T^{2}
53 1−0.261iT−2.80e3T2 1 - 0.261iT - 2.80e3T^{2}
59 1−55.1iT−3.48e3T2 1 - 55.1iT - 3.48e3T^{2}
61 1+104.T+3.72e3T2 1 + 104.T + 3.72e3T^{2}
67 1+64.6T+4.48e3T2 1 + 64.6T + 4.48e3T^{2}
71 1+109.iT−5.04e3T2 1 + 109. iT - 5.04e3T^{2}
73 1+62.9T+5.32e3T2 1 + 62.9T + 5.32e3T^{2}
79 1+19.2T+6.24e3T2 1 + 19.2T + 6.24e3T^{2}
83 1+57.1iT−6.88e3T2 1 + 57.1iT - 6.88e3T^{2}
89 1+103.iT−7.92e3T2 1 + 103. iT - 7.92e3T^{2}
97 1−56.2T+9.40e3T2 1 - 56.2T + 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

−10.13521854203673767999711243284, −8.952584309401744611188088538099, −8.220539540347541771949969042671, −7.32657481064012580910428770213, −6.38475128426038088831270835640, −5.76817829549146263386534900241, −4.44095854577452358297321344436, −3.40822052990353947101598036185, −2.85257732467396786020363462281, −1.74007399070932027535410662092, 0.00177187133409916509943273341, 1.24823129668362277160322138544, 2.62230333367050921800736423188, 4.30899610481086196630257721176, 4.82599089698423367029838233124, 5.33208883630797156036638100533, 6.54322023988547244027904873800, 7.53902427665093741688879802029, 7.888627206351551472182985668332, 9.069616487539839316925166748654

Graph of the ZZ-function along the critical line