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Properties

Label 2-21e2-21.2-c0-0-1
Degree 22
Conductor 441441
Sign 0.675−0.736i0.675 - 0.736i
Analytic cond. 0.2200870.220087
Root an. cond. 0.4691350.469135
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 + 0.707i)2-s + (0.499 + 0.866i)4-s + (−1.22 + 0.707i)11-s + (0.499 − 0.866i)16-s − 2·22-s + (−1.22 − 0.707i)23-s + (−0.5 − 0.866i)25-s + 1.41i·29-s + (1.22 − 0.707i)32-s + (−1.22 − 0.707i)44-s + (−0.999 − 1.73i)46-s − 1.41i·50-s + (1.22 − 0.707i)53-s + (−1.00 + 1.73i)58-s + 0.999·64-s + ⋯
L(s)  = 1  + (1.22 + 0.707i)2-s + (0.499 + 0.866i)4-s + (−1.22 + 0.707i)11-s + (0.499 − 0.866i)16-s − 2·22-s + (−1.22 − 0.707i)23-s + (−0.5 − 0.866i)25-s + 1.41i·29-s + (1.22 − 0.707i)32-s + (−1.22 − 0.707i)44-s + (−0.999 − 1.73i)46-s − 1.41i·50-s + (1.22 − 0.707i)53-s + (−1.00 + 1.73i)58-s + 0.999·64-s + ⋯

Functional equation

Λ(s)=(441s/2 ΓC(s) L(s)=((0.675−0.736i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 - 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(441s/2 ΓC(s) L(s)=((0.675−0.736i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 - 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 441441    =    32⋅723^{2} \cdot 7^{2}
Sign: 0.675−0.736i0.675 - 0.736i
Analytic conductor: 0.2200870.220087
Root analytic conductor: 0.4691350.469135
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ441(422,⋅)\chi_{441} (422, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 441, ( :0), 0.675−0.736i)(2,\ 441,\ (\ :0),\ 0.675 - 0.736i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.4415055181.441505518
L(12)L(\frac12) ≈\approx 1.4415055181.441505518
L(1)L(1) 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
7 1 1
good2 1+(−1.22−0.707i)T+(0.5+0.866i)T2 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2}
5 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
11 1+(1.22−0.707i)T+(0.5−0.866i)T2 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2}
13 1+T2 1 + T^{2}
17 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
19 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
23 1+(1.22+0.707i)T+(0.5+0.866i)T2 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2}
29 1−1.41iT−T2 1 - 1.41iT - T^{2}
31 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
37 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
41 1−T2 1 - T^{2}
43 1+T2 1 + T^{2}
47 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
53 1+(−1.22+0.707i)T+(0.5−0.866i)T2 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2}
59 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
61 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
67 1+(−1−1.73i)T+(−0.5+0.866i)T2 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2}
71 1+1.41iT−T2 1 + 1.41iT - T^{2}
73 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
79 1+(1−1.73i)T+(−0.5−0.866i)T2 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2}
83 1−T2 1 - T^{2}
89 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
97 1+T2 1 + 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

−11.84052208846762375279621408714, −10.49103574622163080226566462596, −9.864992801402636931402749329696, −8.427010539387864622455064897460, −7.51647339343988795042853942915, −6.65004984613259944318897430329, −5.63783728162440250647776145816, −4.83744785579779234334522478818, −3.86931961490569081585999189777, −2.47891726023275405362271729690, 2.14017190261486800664524466824, 3.25394871834383751516281348352, 4.24701759009354368989110402291, 5.41790831006616825823780459578, 5.99673600071139469592678931300, 7.59703851389044336707338211636, 8.385762160058102779869942834926, 9.750816994845927112963944443200, 10.63406120038908678855810746401, 11.43467133037809501503769290651

Graph of the ZZ-function along the critical line