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Properties

Label 2-1458-3.2-c2-0-24
Degree 22
Conductor 14581458
Sign 11
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 − 0.589i·5-s − 3.15·7-s + 2.82i·8-s − 0.833·10-s − 9.38i·11-s + 12.2·13-s + 4.45i·14-s + 4.00·16-s + 28.3i·17-s − 22.4·19-s + 1.17i·20-s − 13.2·22-s − 10.0i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.500·4-s − 0.117i·5-s − 0.450·7-s + 0.353i·8-s − 0.0833·10-s − 0.853i·11-s + 0.939·13-s + 0.318i·14-s + 0.250·16-s + 1.66i·17-s − 1.17·19-s + 0.0589i·20-s − 0.603·22-s − 0.437i·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: 11
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 1.4388465021.438846502
L(12)L(\frac12) ≈\approx 1.4388465021.438846502
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+0.589iT−25T2 1 + 0.589iT - 25T^{2}
7 1+3.15T+49T2 1 + 3.15T + 49T^{2}
11 1+9.38iT−121T2 1 + 9.38iT - 121T^{2}
13 1−12.2T+169T2 1 - 12.2T + 169T^{2}
17 1−28.3iT−289T2 1 - 28.3iT - 289T^{2}
19 1+22.4T+361T2 1 + 22.4T + 361T^{2}
23 1+10.0iT−529T2 1 + 10.0iT - 529T^{2}
29 1−24.0iT−841T2 1 - 24.0iT - 841T^{2}
31 1+45.4T+961T2 1 + 45.4T + 961T^{2}
37 1−31.5T+1.36e3T2 1 - 31.5T + 1.36e3T^{2}
41 1−70.8iT−1.68e3T2 1 - 70.8iT - 1.68e3T^{2}
43 1+14.5T+1.84e3T2 1 + 14.5T + 1.84e3T^{2}
47 1+55.3iT−2.20e3T2 1 + 55.3iT - 2.20e3T^{2}
53 1−25.4iT−2.80e3T2 1 - 25.4iT - 2.80e3T^{2}
59 1+28.6iT−3.48e3T2 1 + 28.6iT - 3.48e3T^{2}
61 1−113.T+3.72e3T2 1 - 113.T + 3.72e3T^{2}
67 1−50.0T+4.48e3T2 1 - 50.0T + 4.48e3T^{2}
71 1−8.77iT−5.04e3T2 1 - 8.77iT - 5.04e3T^{2}
73 1−23.4T+5.32e3T2 1 - 23.4T + 5.32e3T^{2}
79 1−132.T+6.24e3T2 1 - 132.T + 6.24e3T^{2}
83 1−67.1iT−6.88e3T2 1 - 67.1iT - 6.88e3T^{2}
89 1−72.7iT−7.92e3T2 1 - 72.7iT - 7.92e3T^{2}
97 1−157.T+9.40e3T2 1 - 157.T + 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

−9.302705205065056171966315989244, −8.526800945680272487566451660429, −8.168596304264759261091712993550, −6.63441501924639036909862771740, −6.12044057673371334387601414826, −5.06956105860214646569182316522, −3.90116021395459799261367875647, −3.37516057797265839630333079647, −2.08756476468498690097528654399, −0.940938014598044732413071589097, 0.50619688218407664804053777712, 2.14380287870122546436372983843, 3.40011536886970913730143861660, 4.36577531628789120151368123607, 5.24305744217178257583950306486, 6.17841647564885302023106813805, 6.93480516309525032069332354432, 7.52181051194335801009442542334, 8.550255336818476996399552360242, 9.255124765875452977460008835492

Graph of the ZZ-function along the critical line