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Properties

Label 2-21e2-49.8-c1-0-7
Degree $2$
Conductor $441$
Sign $-0.878 - 0.478i$
Analytic cond. $3.52140$
Root an. cond. $1.87654$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.57 + 1.97i)2-s + (−0.973 + 4.26i)4-s + (0.136 − 0.0655i)5-s + (0.357 + 2.62i)7-s + (−5.39 + 2.60i)8-s + (0.343 + 0.165i)10-s + (−2.31 − 2.89i)11-s + (1.07 + 1.34i)13-s + (−4.61 + 4.83i)14-s + (−5.75 − 2.76i)16-s + (0.969 + 4.24i)17-s + 1.41·19-s + (0.147 + 0.644i)20-s + (2.08 − 9.12i)22-s + (1.39 − 6.12i)23-s + ⋯
L(s)  = 1  + (1.11 + 1.39i)2-s + (−0.486 + 2.13i)4-s + (0.0609 − 0.0293i)5-s + (0.134 + 0.990i)7-s + (−1.90 + 0.919i)8-s + (0.108 + 0.0523i)10-s + (−0.696 − 0.873i)11-s + (0.297 + 0.373i)13-s + (−1.23 + 1.29i)14-s + (−1.43 − 0.692i)16-s + (0.235 + 1.03i)17-s + 0.323·19-s + (0.0329 + 0.144i)20-s + (0.443 − 1.94i)22-s + (0.291 − 1.27i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.878 - 0.478i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.878 - 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.878 - 0.478i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.878 - 0.478i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.573861 + 2.25384i\)
\(L(\frac12)\) \(\approx\) \(0.573861 + 2.25384i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.357 - 2.62i)T \)
good2 \( 1 + (-1.57 - 1.97i)T + (-0.445 + 1.94i)T^{2} \)
5 \( 1 + (-0.136 + 0.0655i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (2.31 + 2.89i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (-1.07 - 1.34i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + (-0.969 - 4.24i)T + (-15.3 + 7.37i)T^{2} \)
19 \( 1 - 1.41T + 19T^{2} \)
23 \( 1 + (-1.39 + 6.12i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (2.01 + 8.82i)T + (-26.1 + 12.5i)T^{2} \)
31 \( 1 - 10.9T + 31T^{2} \)
37 \( 1 + (0.234 + 1.02i)T + (-33.3 + 16.0i)T^{2} \)
41 \( 1 + (-5.10 + 2.45i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (3.23 + 1.55i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (-4.37 - 5.48i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (1.62 - 7.11i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 + (-1.25 - 0.602i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (3.21 + 14.0i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 - 8.12T + 67T^{2} \)
71 \( 1 + (-0.311 + 1.36i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-1.96 + 2.46i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 - 12.4T + 79T^{2} \)
83 \( 1 + (-0.268 + 0.337i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (-7.24 + 9.07i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 + 0.497T + 97T^{2} \)
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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

−11.87788383165708441805742056844, −10.79426923697785059835131846845, −9.339361387989871874634858354965, −8.262092129784618946201549943936, −7.902910400480977028278587710400, −6.39984075146360825888914778918, −5.95216715017678264616258427666, −5.03890646043497381545438308598, −3.94212387107028834253577096226, −2.67637083926064379603874927719, 1.16071083470690150350247574902, 2.62870114738872905068107434264, 3.66654606643924143000773729763, 4.72726013861004471487853351276, 5.42281842770789070967679322380, 6.87596527078819974705278413947, 7.923102190956306807115531675030, 9.573060266520888220809531120703, 10.12661448077273384643262906734, 10.87767404368288910859183070297

Graph of the $Z$-function along the critical line