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Properties

Label 2-21e2-7.5-c0-0-0
Degree 22
Conductor 441441
Sign 0.832−0.553i0.832 - 0.553i
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  + (0.5 + 0.866i)4-s + (−0.499 + 0.866i)16-s + (−0.5 − 0.866i)25-s + (1 − 1.73i)37-s − 2·43-s − 0.999·64-s + (−1 − 1.73i)67-s + (−1 + 1.73i)79-s + (0.499 − 0.866i)100-s + (1 + 1.73i)109-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)4-s + (−0.499 + 0.866i)16-s + (−0.5 − 0.866i)25-s + (1 − 1.73i)37-s − 2·43-s − 0.999·64-s + (−1 − 1.73i)67-s + (−1 + 1.73i)79-s + (0.499 − 0.866i)100-s + (1 + 1.73i)109-s + ⋯

Functional equation

Λ(s)=(441s/2 ΓC(s) L(s)=((0.832−0.553i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(441s/2 ΓC(s) L(s)=((0.832−0.553i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 441441    =    32⋅723^{2} \cdot 7^{2}
Sign: 0.832−0.553i0.832 - 0.553i
Analytic conductor: 0.2200870.220087
Root analytic conductor: 0.4691350.469135
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ441(19,⋅)\chi_{441} (19, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 441, ( :0), 0.832−0.553i)(2,\ 441,\ (\ :0),\ 0.832 - 0.553i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.92939903990.9293990399
L(12)L(\frac12) ≈\approx 0.92939903990.9293990399
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+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
5 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
11 1+(−0.5+0.866i)T2 1 + (-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+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
29 1+T2 1 + T^{2}
31 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
37 1+(−1+1.73i)T+(−0.5−0.866i)T2 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2}
41 1−T2 1 - T^{2}
43 1+2T+T2 1 + 2T + T^{2}
47 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
53 1+(−0.5+0.866i)T2 1 + (-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+T2 1 + 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

−11.49438970359066473817352013659, −10.67421072164152299370820510375, −9.609722008000977215470084698991, −8.549787287379746596472714024378, −7.78623839517945642711281070580, −6.87834066262158608905266997488, −5.90646054431188370473342496616, −4.48029662965645927091740316497, −3.39923831260526179066932882330, −2.16617937203495717213378069952, 1.60038641080423647027116313974, 3.05249497320041956745119721480, 4.61122384632602001394216453963, 5.63467893600201756616122681466, 6.52676742060248642637602831359, 7.45134713216296716150282292034, 8.591224729354076119968779401602, 9.698700915356889802984301980440, 10.26463573503206159132979439772, 11.33626292852082760209850477605

Graph of the ZZ-function along the critical line