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Properties

Label 2-21e2-7.5-c0-0-0
Degree $2$
Conductor $441$
Sign $0.832 - 0.553i$
Analytic cond. $0.220087$
Root an. cond. $0.469135$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.832 - 0.553i$
Analytic conductor: \(0.220087\)
Root analytic conductor: \(0.469135\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :0),\ 0.832 - 0.553i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9293990399\)
\(L(\frac12)\) \(\approx\) \(0.9293990399\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 2T + T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line