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Properties

Label 2-21e2-63.13-c2-0-42
Degree 22
Conductor 441441
Sign 0.302−0.953i0.302 - 0.953i
Analytic cond. 12.016312.0163
Root an. cond. 3.466463.46646
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  + (1.41 + 2.45i)2-s + (2.70 − 1.30i)3-s + (−2.01 + 3.49i)4-s + (2.07 + 1.19i)5-s + (7.02 + 4.78i)6-s − 0.0884·8-s + (5.60 − 7.04i)9-s + 6.79i·10-s + (−5.69 − 9.86i)11-s + (−0.896 + 12.0i)12-s + (13.4 + 7.77i)13-s + (7.17 + 0.533i)15-s + (7.93 + 13.7i)16-s + 12.1i·17-s + (25.2 + 3.77i)18-s + 16.0i·19-s + ⋯
L(s)  = 1  + (0.708 + 1.22i)2-s + (0.900 − 0.434i)3-s + (−0.503 + 0.872i)4-s + (0.415 + 0.239i)5-s + (1.17 + 0.797i)6-s − 0.0110·8-s + (0.622 − 0.782i)9-s + 0.679i·10-s + (−0.517 − 0.896i)11-s + (−0.0747 + 1.00i)12-s + (1.03 + 0.597i)13-s + (0.478 + 0.0355i)15-s + (0.496 + 0.859i)16-s + 0.711i·17-s + (1.40 + 0.209i)18-s + 0.842i·19-s + ⋯

Functional equation

Λ(s)=(441s/2 ΓC(s) L(s)=((0.302−0.953i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.302 - 0.953i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(441s/2 ΓC(s+1) L(s)=((0.302−0.953i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.302 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 441441    =    32⋅723^{2} \cdot 7^{2}
Sign: 0.302−0.953i0.302 - 0.953i
Analytic conductor: 12.016312.0163
Root analytic conductor: 3.466463.46646
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ441(391,⋅)\chi_{441} (391, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 441, ( :1), 0.302−0.953i)(2,\ 441,\ (\ :1),\ 0.302 - 0.953i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 3.07281+2.24845i3.07281 + 2.24845i
L(12)L(\frac12) ≈\approx 3.07281+2.24845i3.07281 + 2.24845i
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)
bad3 1+(−2.70+1.30i)T 1 + (-2.70 + 1.30i)T
7 1 1
good2 1+(−1.41−2.45i)T+(−2+3.46i)T2 1 + (-1.41 - 2.45i)T + (-2 + 3.46i)T^{2}
5 1+(−2.07−1.19i)T+(12.5+21.6i)T2 1 + (-2.07 - 1.19i)T + (12.5 + 21.6i)T^{2}
11 1+(5.69+9.86i)T+(−60.5+104.i)T2 1 + (5.69 + 9.86i)T + (-60.5 + 104. i)T^{2}
13 1+(−13.4−7.77i)T+(84.5+146.i)T2 1 + (-13.4 - 7.77i)T + (84.5 + 146. i)T^{2}
17 1−12.1iT−289T2 1 - 12.1iT - 289T^{2}
19 1−16.0iT−361T2 1 - 16.0iT - 361T^{2}
23 1+(−8.02+13.9i)T+(−264.5−458.i)T2 1 + (-8.02 + 13.9i)T + (-264.5 - 458. i)T^{2}
29 1+(−16.2−28.1i)T+(−420.5+728.i)T2 1 + (-16.2 - 28.1i)T + (-420.5 + 728. i)T^{2}
31 1+(36.7+21.1i)T+(480.5+832.i)T2 1 + (36.7 + 21.1i)T + (480.5 + 832. i)T^{2}
37 1−7.22T+1.36e3T2 1 - 7.22T + 1.36e3T^{2}
41 1+(−7.55−4.36i)T+(840.5+1.45e3i)T2 1 + (-7.55 - 4.36i)T + (840.5 + 1.45e3i)T^{2}
43 1+(22.2+38.4i)T+(−924.5+1.60e3i)T2 1 + (22.2 + 38.4i)T + (-924.5 + 1.60e3i)T^{2}
47 1+(22.8−13.1i)T+(1.10e3−1.91e3i)T2 1 + (22.8 - 13.1i)T + (1.10e3 - 1.91e3i)T^{2}
53 1+68.1T+2.80e3T2 1 + 68.1T + 2.80e3T^{2}
59 1+(−82.4−47.5i)T+(1.74e3+3.01e3i)T2 1 + (-82.4 - 47.5i)T + (1.74e3 + 3.01e3i)T^{2}
61 1+(42.8−24.7i)T+(1.86e3−3.22e3i)T2 1 + (42.8 - 24.7i)T + (1.86e3 - 3.22e3i)T^{2}
67 1+(−40.9+70.9i)T+(−2.24e3−3.88e3i)T2 1 + (-40.9 + 70.9i)T + (-2.24e3 - 3.88e3i)T^{2}
71 1+112.T+5.04e3T2 1 + 112.T + 5.04e3T^{2}
73 1−67.9iT−5.32e3T2 1 - 67.9iT - 5.32e3T^{2}
79 1+(68.4+118.i)T+(−3.12e3+5.40e3i)T2 1 + (68.4 + 118. i)T + (-3.12e3 + 5.40e3i)T^{2}
83 1+(18.8−10.8i)T+(3.44e3−5.96e3i)T2 1 + (18.8 - 10.8i)T + (3.44e3 - 5.96e3i)T^{2}
89 1+108.iT−7.92e3T2 1 + 108. iT - 7.92e3T^{2}
97 1+(96.5−55.7i)T+(4.70e3−8.14e3i)T2 1 + (96.5 - 55.7i)T + (4.70e3 - 8.14e3i)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

−11.02099655328373562044998858716, −10.12114978969033855082332559693, −8.741222486832519558526335186377, −8.268074982843073120439423037058, −7.26756120050145131593110217968, −6.32353155094211230873963128525, −5.78015041109981685114066441137, −4.27263639202951085382490172914, −3.29453002358147517299257623805, −1.68044317597475058201896271959, 1.51347423832403732507719484279, 2.63384500531772754545526602012, 3.51973381346497445243283742927, 4.63361652640927256018952979912, 5.39359861589545021799754131725, 7.16676003003977867780031903544, 8.141353057410999501963171570164, 9.344859633412766607775606692072, 9.884844084198086543350057732857, 10.81363338838315319374693259147

Graph of the ZZ-function along the critical line