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Properties

Label 2-1458-3.2-c2-0-59
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 + 3.04i·5-s + 8.22·7-s + 2.82i·8-s + 4.30·10-s + 6.55i·11-s − 18.5·13-s − 11.6i·14-s + 4.00·16-s − 13.1i·17-s − 34.8·19-s − 6.08i·20-s + 9.26·22-s + 3.31i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.500·4-s + 0.608i·5-s + 1.17·7-s + 0.353i·8-s + 0.430·10-s + 0.595i·11-s − 1.42·13-s − 0.831i·14-s + 0.250·16-s − 0.776i·17-s − 1.83·19-s − 0.304i·20-s + 0.421·22-s + 0.144i·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 0.43217589430.4321758943
L(12)L(\frac12) ≈\approx 0.43217589430.4321758943
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 1−3.04iT−25T2 1 - 3.04iT - 25T^{2}
7 1−8.22T+49T2 1 - 8.22T + 49T^{2}
11 1−6.55iT−121T2 1 - 6.55iT - 121T^{2}
13 1+18.5T+169T2 1 + 18.5T + 169T^{2}
17 1+13.1iT−289T2 1 + 13.1iT - 289T^{2}
19 1+34.8T+361T2 1 + 34.8T + 361T^{2}
23 1−3.31iT−529T2 1 - 3.31iT - 529T^{2}
29 1−28.4iT−841T2 1 - 28.4iT - 841T^{2}
31 1+13.8T+961T2 1 + 13.8T + 961T^{2}
37 1−6.66T+1.36e3T2 1 - 6.66T + 1.36e3T^{2}
41 1+56.9iT−1.68e3T2 1 + 56.9iT - 1.68e3T^{2}
43 1+32.7T+1.84e3T2 1 + 32.7T + 1.84e3T^{2}
47 1+53.6iT−2.20e3T2 1 + 53.6iT - 2.20e3T^{2}
53 1+98.5iT−2.80e3T2 1 + 98.5iT - 2.80e3T^{2}
59 1+102.iT−3.48e3T2 1 + 102. iT - 3.48e3T^{2}
61 1+7.18T+3.72e3T2 1 + 7.18T + 3.72e3T^{2}
67 1−30.0T+4.48e3T2 1 - 30.0T + 4.48e3T^{2}
71 1−9.77iT−5.04e3T2 1 - 9.77iT - 5.04e3T^{2}
73 1+129.T+5.32e3T2 1 + 129.T + 5.32e3T^{2}
79 1−102.T+6.24e3T2 1 - 102.T + 6.24e3T^{2}
83 1+108.iT−6.88e3T2 1 + 108. iT - 6.88e3T^{2}
89 1−23.2iT−7.92e3T2 1 - 23.2iT - 7.92e3T^{2}
97 1+51.2T+9.40e3T2 1 + 51.2T + 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

−8.989304185714563869618022179404, −8.229510572182436498316999790566, −7.30136736667982916161940021565, −6.69143770122737018953986153416, −5.12945141888869012262599643407, −4.81914291119925388077321261090, −3.67776485987459971582688250450, −2.43101580577122059586867138338, −1.85386503187915005560199026917, −0.11649204139320019006131753565, 1.36414661276739584507652065634, 2.60845675706147207467311722580, 4.31264985553288243098981225822, 4.61898482845015389362614061402, 5.62542956864327965995487684870, 6.41063733203571713624297772311, 7.46880362443392254845553860167, 8.189772059903430396137632117422, 8.632002833616108250148537362191, 9.518092753668421004187503305285

Graph of the ZZ-function along the critical line