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Properties

Label 2-22e2-121.100-c1-0-3
Degree 22
Conductor 484484
Sign 0.374−0.927i0.374 - 0.927i
Analytic cond. 3.864753.86475
Root an. cond. 1.965891.96589
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  − 0.390·3-s + (1.96 − 0.576i)5-s + (−0.526 + 3.65i)7-s − 2.84·9-s + (2.68 + 1.94i)11-s + (−4.47 + 5.16i)13-s + (−0.767 + 0.225i)15-s + (4.70 − 3.02i)17-s + (3.38 + 2.17i)19-s + (0.205 − 1.42i)21-s + (−0.357 − 2.48i)23-s + (−0.680 + 0.437i)25-s + 2.28·27-s + (−4.22 − 2.71i)29-s + (4.81 + 5.55i)31-s + ⋯
L(s)  = 1  − 0.225·3-s + (0.878 − 0.257i)5-s + (−0.198 + 1.38i)7-s − 0.949·9-s + (0.809 + 0.586i)11-s + (−1.24 + 1.43i)13-s + (−0.198 + 0.0581i)15-s + (1.14 − 0.732i)17-s + (0.777 + 0.499i)19-s + (0.0448 − 0.311i)21-s + (−0.0745 − 0.518i)23-s + (−0.136 + 0.0874i)25-s + 0.439·27-s + (−0.785 − 0.504i)29-s + (0.864 + 0.997i)31-s + ⋯

Functional equation

Λ(s)=(484s/2 ΓC(s) L(s)=((0.374−0.927i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(484s/2 ΓC(s+1/2) L(s)=((0.374−0.927i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 484484    =    22⋅1122^{2} \cdot 11^{2}
Sign: 0.374−0.927i0.374 - 0.927i
Analytic conductor: 3.864753.86475
Root analytic conductor: 1.965891.96589
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ484(221,⋅)\chi_{484} (221, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 484, ( :1/2), 0.374−0.927i)(2,\ 484,\ (\ :1/2),\ 0.374 - 0.927i)

Particular Values

L(1)L(1) ≈\approx 1.10068+0.742299i1.10068 + 0.742299i
L(12)L(\frac12) ≈\approx 1.10068+0.742299i1.10068 + 0.742299i
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)
bad2 1 1
11 1+(−2.68−1.94i)T 1 + (-2.68 - 1.94i)T
good3 1+0.390T+3T2 1 + 0.390T + 3T^{2}
5 1+(−1.96+0.576i)T+(4.20−2.70i)T2 1 + (-1.96 + 0.576i)T + (4.20 - 2.70i)T^{2}
7 1+(0.526−3.65i)T+(−6.71−1.97i)T2 1 + (0.526 - 3.65i)T + (-6.71 - 1.97i)T^{2}
13 1+(4.47−5.16i)T+(−1.85−12.8i)T2 1 + (4.47 - 5.16i)T + (-1.85 - 12.8i)T^{2}
17 1+(−4.70+3.02i)T+(7.06−15.4i)T2 1 + (-4.70 + 3.02i)T + (7.06 - 15.4i)T^{2}
19 1+(−3.38−2.17i)T+(7.89+17.2i)T2 1 + (-3.38 - 2.17i)T + (7.89 + 17.2i)T^{2}
23 1+(0.357+2.48i)T+(−22.0+6.47i)T2 1 + (0.357 + 2.48i)T + (-22.0 + 6.47i)T^{2}
29 1+(4.22+2.71i)T+(12.0+26.3i)T2 1 + (4.22 + 2.71i)T + (12.0 + 26.3i)T^{2}
31 1+(−4.81−5.55i)T+(−4.41+30.6i)T2 1 + (-4.81 - 5.55i)T + (-4.41 + 30.6i)T^{2}
37 1+(2.20+2.54i)T+(−5.26+36.6i)T2 1 + (2.20 + 2.54i)T + (-5.26 + 36.6i)T^{2}
41 1+(−5.18−11.3i)T+(−26.8+30.9i)T2 1 + (-5.18 - 11.3i)T + (-26.8 + 30.9i)T^{2}
43 1+(−5.07−1.49i)T+(36.1+23.2i)T2 1 + (-5.07 - 1.49i)T + (36.1 + 23.2i)T^{2}
47 1+(−1.72+3.76i)T+(−30.7−35.5i)T2 1 + (-1.72 + 3.76i)T + (-30.7 - 35.5i)T^{2}
53 1+(−0.659+4.58i)T+(−50.8−14.9i)T2 1 + (-0.659 + 4.58i)T + (-50.8 - 14.9i)T^{2}
59 1+(0.325−0.712i)T+(−38.6−44.5i)T2 1 + (0.325 - 0.712i)T + (-38.6 - 44.5i)T^{2}
61 1+(0.440−0.963i)T+(−39.9−46.1i)T2 1 + (0.440 - 0.963i)T + (-39.9 - 46.1i)T^{2}
67 1+(2.04+4.48i)T+(−43.8+50.6i)T2 1 + (2.04 + 4.48i)T + (-43.8 + 50.6i)T^{2}
71 1+(1.87+1.20i)T+(29.4+64.5i)T2 1 + (1.87 + 1.20i)T + (29.4 + 64.5i)T^{2}
73 1+(2.25+15.6i)T+(−70.0+20.5i)T2 1 + (2.25 + 15.6i)T + (-70.0 + 20.5i)T^{2}
79 1+(6.55−1.92i)T+(66.4−42.7i)T2 1 + (6.55 - 1.92i)T + (66.4 - 42.7i)T^{2}
83 1+(−0.413+2.87i)T+(−79.6−23.3i)T2 1 + (-0.413 + 2.87i)T + (-79.6 - 23.3i)T^{2}
89 1+(3.21−2.06i)T+(36.9−80.9i)T2 1 + (3.21 - 2.06i)T + (36.9 - 80.9i)T^{2}
97 1+(2.35+0.692i)T+(81.6+52.4i)T2 1 + (2.35 + 0.692i)T + (81.6 + 52.4i)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

−11.50919218105039568926820259086, −9.828837307569495785721472315872, −9.489088841206676445463040870316, −8.763875130011546432569479455848, −7.43939116418482654811475253904, −6.28289687200564523982515235929, −5.59000317098783256511337178100, −4.71119437683156271188138303631, −2.91749668816696841375754615344, −1.86640966862029136016733709865, 0.853900440263696063693794559654, 2.78226991980312964230984972624, 3.83274220963884978498180902084, 5.42653450896760999796882670734, 5.95174104346691633036343869908, 7.20481490593990754083549698778, 7.913628320203575000502148762334, 9.241837725504973643569649819229, 10.09051521587906351122520873878, 10.60712027572204728995067369422

Graph of the ZZ-function along the critical line