Properties

Label 2-1458-9.7-c1-0-22
Degree 22
Conductor 14581458
Sign 0.5+0.866i0.5 + 0.866i
Analytic cond. 11.642111.6421
Root an. cond. 3.412063.41206
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (1.53 + 2.66i)5-s + (0.127 − 0.220i)7-s − 0.999·8-s + 3.07·10-s + (1.72 − 2.99i)11-s + (−2.81 − 4.88i)13-s + (−0.127 − 0.220i)14-s + (−0.5 + 0.866i)16-s + 5.01·17-s − 0.904·19-s + (1.53 − 2.66i)20-s + (−1.72 − 2.99i)22-s + (−2.82 − 4.89i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.688 + 1.19i)5-s + (0.0480 − 0.0832i)7-s − 0.353·8-s + 0.973·10-s + (0.520 − 0.901i)11-s + (−0.782 − 1.35i)13-s + (−0.0339 − 0.0588i)14-s + (−0.125 + 0.216i)16-s + 1.21·17-s − 0.207·19-s + (0.344 − 0.596i)20-s + (−0.368 − 0.637i)22-s + (−0.589 − 1.02i)23-s + ⋯

Functional equation

Λ(s)=(1458s/2ΓC(s)L(s)=((0.5+0.866i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1458s/2ΓC(s+1/2)L(s)=((0.5+0.866i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14581458    =    2362 \cdot 3^{6}
Sign: 0.5+0.866i0.5 + 0.866i
Analytic conductor: 11.642111.6421
Root analytic conductor: 3.412063.41206
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1458(487,)\chi_{1458} (487, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1458, ( :1/2), 0.5+0.866i)(2,\ 1458,\ (\ :1/2),\ 0.5 + 0.866i)

Particular Values

L(1)L(1) \approx 2.2685220332.268522033
L(12)L(\frac12) \approx 2.2685220332.268522033
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(0.5+0.866i)T 1 + (-0.5 + 0.866i)T
3 1 1
good5 1+(1.532.66i)T+(2.5+4.33i)T2 1 + (-1.53 - 2.66i)T + (-2.5 + 4.33i)T^{2}
7 1+(0.127+0.220i)T+(3.56.06i)T2 1 + (-0.127 + 0.220i)T + (-3.5 - 6.06i)T^{2}
11 1+(1.72+2.99i)T+(5.59.52i)T2 1 + (-1.72 + 2.99i)T + (-5.5 - 9.52i)T^{2}
13 1+(2.81+4.88i)T+(6.5+11.2i)T2 1 + (2.81 + 4.88i)T + (-6.5 + 11.2i)T^{2}
17 15.01T+17T2 1 - 5.01T + 17T^{2}
19 1+0.904T+19T2 1 + 0.904T + 19T^{2}
23 1+(2.82+4.89i)T+(11.5+19.9i)T2 1 + (2.82 + 4.89i)T + (-11.5 + 19.9i)T^{2}
29 1+(0.883+1.53i)T+(14.525.1i)T2 1 + (-0.883 + 1.53i)T + (-14.5 - 25.1i)T^{2}
31 1+(5.329.23i)T+(15.5+26.8i)T2 1 + (-5.32 - 9.23i)T + (-15.5 + 26.8i)T^{2}
37 18.83T+37T2 1 - 8.83T + 37T^{2}
41 1+(2.854.94i)T+(20.5+35.5i)T2 1 + (-2.85 - 4.94i)T + (-20.5 + 35.5i)T^{2}
43 1+(4.28+7.41i)T+(21.537.2i)T2 1 + (-4.28 + 7.41i)T + (-21.5 - 37.2i)T^{2}
47 1+(6.26+10.8i)T+(23.540.7i)T2 1 + (-6.26 + 10.8i)T + (-23.5 - 40.7i)T^{2}
53 12.65T+53T2 1 - 2.65T + 53T^{2}
59 1+(1.85+3.20i)T+(29.5+51.0i)T2 1 + (1.85 + 3.20i)T + (-29.5 + 51.0i)T^{2}
61 1+(1.88+3.27i)T+(30.552.8i)T2 1 + (-1.88 + 3.27i)T + (-30.5 - 52.8i)T^{2}
67 1+(3.796.56i)T+(33.5+58.0i)T2 1 + (-3.79 - 6.56i)T + (-33.5 + 58.0i)T^{2}
71 1+8.42T+71T2 1 + 8.42T + 71T^{2}
73 1+6.99T+73T2 1 + 6.99T + 73T^{2}
79 1+(6.5511.3i)T+(39.568.4i)T2 1 + (6.55 - 11.3i)T + (-39.5 - 68.4i)T^{2}
83 1+(3.46+5.99i)T+(41.571.8i)T2 1 + (-3.46 + 5.99i)T + (-41.5 - 71.8i)T^{2}
89 1+6.80T+89T2 1 + 6.80T + 89T^{2}
97 1+(4.317.47i)T+(48.584.0i)T2 1 + (4.31 - 7.47i)T + (-48.5 - 84.0i)T^{2}
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   L(s)=p j=12(1αj,pps)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

−9.765108458968073389315886171109, −8.644990710497560818884797321098, −7.78115359346841863424083299701, −6.76275478859783144101673745564, −5.97620075871321614912365304132, −5.35774571345324787599632121839, −4.06985730054284845411436447960, −2.99370849820331393747076848592, −2.56168284459775077006599120658, −0.931829821947786026922724150740, 1.31322012294842728329271208419, 2.44394221907385955318552184250, 4.17740060733371813208804250991, 4.54092932061845182690857185012, 5.60314515554228153481570924626, 6.16638970505204570958860022592, 7.31123012067473401153064700364, 7.86902969151416194173426655081, 9.017746678126506841454626902291, 9.487393057378062789012963117077

Graph of the ZZ-function along the critical line