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Properties

Label 2-297-9.4-c1-0-6
Degree 22
Conductor 297297
Sign −0.939+0.342i-0.939 + 0.342i
Analytic cond. 2.371552.37155
Root an. cond. 1.539981.53998
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  + (−0.939 − 1.62i)2-s + (−0.766 + 1.32i)4-s + (1.43 − 2.49i)5-s + (0.326 + 0.565i)7-s − 0.879·8-s − 5.41·10-s + (−0.5 − 0.866i)11-s + (3.37 − 5.85i)13-s + (0.613 − 1.06i)14-s + (2.35 + 4.08i)16-s − 0.184·17-s − 5.22·19-s + (2.20 + 3.82i)20-s + (−0.939 + 1.62i)22-s + (−1.59 + 2.75i)23-s + ⋯
L(s)  = 1  + (−0.664 − 1.15i)2-s + (−0.383 + 0.663i)4-s + (0.643 − 1.11i)5-s + (0.123 + 0.213i)7-s − 0.310·8-s − 1.71·10-s + (−0.150 − 0.261i)11-s + (0.937 − 1.62i)13-s + (0.163 − 0.283i)14-s + (0.589 + 1.02i)16-s − 0.0448·17-s − 1.19·19-s + (0.493 + 0.854i)20-s + (−0.200 + 0.347i)22-s + (−0.332 + 0.575i)23-s + ⋯

Functional equation

Λ(s)=(297s/2 ΓC(s) L(s)=((−0.939+0.342i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(297s/2 ΓC(s+1/2) L(s)=((−0.939+0.342i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 297297    =    33⋅113^{3} \cdot 11
Sign: −0.939+0.342i-0.939 + 0.342i
Analytic conductor: 2.371552.37155
Root analytic conductor: 1.539981.53998
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ297(199,⋅)\chi_{297} (199, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 297, ( :1/2), −0.939+0.342i)(2,\ 297,\ (\ :1/2),\ -0.939 + 0.342i)

Particular Values

L(1)L(1) ≈\approx 0.162391−0.920966i0.162391 - 0.920966i
L(12)L(\frac12) ≈\approx 0.162391−0.920966i0.162391 - 0.920966i
L(32)L(\frac{3}{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)
bad3 1 1
11 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
good2 1+(0.939+1.62i)T+(−1+1.73i)T2 1 + (0.939 + 1.62i)T + (-1 + 1.73i)T^{2}
5 1+(−1.43+2.49i)T+(−2.5−4.33i)T2 1 + (-1.43 + 2.49i)T + (-2.5 - 4.33i)T^{2}
7 1+(−0.326−0.565i)T+(−3.5+6.06i)T2 1 + (-0.326 - 0.565i)T + (-3.5 + 6.06i)T^{2}
13 1+(−3.37+5.85i)T+(−6.5−11.2i)T2 1 + (-3.37 + 5.85i)T + (-6.5 - 11.2i)T^{2}
17 1+0.184T+17T2 1 + 0.184T + 17T^{2}
19 1+5.22T+19T2 1 + 5.22T + 19T^{2}
23 1+(1.59−2.75i)T+(−11.5−19.9i)T2 1 + (1.59 - 2.75i)T + (-11.5 - 19.9i)T^{2}
29 1+(2.01+3.48i)T+(−14.5+25.1i)T2 1 + (2.01 + 3.48i)T + (-14.5 + 25.1i)T^{2}
31 1+(0.553−0.957i)T+(−15.5−26.8i)T2 1 + (0.553 - 0.957i)T + (-15.5 - 26.8i)T^{2}
37 1−0.106T+37T2 1 - 0.106T + 37T^{2}
41 1+(2.80−4.86i)T+(−20.5−35.5i)T2 1 + (2.80 - 4.86i)T + (-20.5 - 35.5i)T^{2}
43 1+(1.92+3.34i)T+(−21.5+37.2i)T2 1 + (1.92 + 3.34i)T + (-21.5 + 37.2i)T^{2}
47 1+(−6.00−10.3i)T+(−23.5+40.7i)T2 1 + (-6.00 - 10.3i)T + (-23.5 + 40.7i)T^{2}
53 1−10.0T+53T2 1 - 10.0T + 53T^{2}
59 1+(−5.27+9.14i)T+(−29.5−51.0i)T2 1 + (-5.27 + 9.14i)T + (-29.5 - 51.0i)T^{2}
61 1+(−3.67−6.36i)T+(−30.5+52.8i)T2 1 + (-3.67 - 6.36i)T + (-30.5 + 52.8i)T^{2}
67 1+(−5.90+10.2i)T+(−33.5−58.0i)T2 1 + (-5.90 + 10.2i)T + (-33.5 - 58.0i)T^{2}
71 1−2.47T+71T2 1 - 2.47T + 71T^{2}
73 1−10.4T+73T2 1 - 10.4T + 73T^{2}
79 1+(−0.733−1.27i)T+(−39.5+68.4i)T2 1 + (-0.733 - 1.27i)T + (-39.5 + 68.4i)T^{2}
83 1+(0.520+0.902i)T+(−41.5+71.8i)T2 1 + (0.520 + 0.902i)T + (-41.5 + 71.8i)T^{2}
89 1−3.01T+89T2 1 - 3.01T + 89T^{2}
97 1+(−2.86−4.97i)T+(−48.5+84.0i)T2 1 + (-2.86 - 4.97i)T + (-48.5 + 84.0i)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

−11.15163333380977222181912900135, −10.42647146572784329545542944206, −9.573707659301433004451754353606, −8.661215908162822105077658541302, −8.122907256293427768156238648713, −6.11004028884001231456362969889, −5.32064532355896605075344769924, −3.67436588405866149105852126516, −2.22081221963308618881918913969, −0.884530217831679671929488702766, 2.24683531682321304980551133672, 3.98648796847516815450578457915, 5.70144495798354710881880417766, 6.70535181082041685499618110259, 6.97548671456027712794225657824, 8.372300765466276839193214309410, 9.103120815496979919447757794946, 10.17239154420141317267422700749, 10.96180165205733018514587104901, 12.05163482902305795797138903105

Graph of the ZZ-function along the critical line