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Properties

Label 2-21e2-147.131-c1-0-5
Degree $2$
Conductor $441$
Sign $0.239 - 0.971i$
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  + (1.50 − 0.588i)2-s + (0.437 − 0.406i)4-s + (−3.18 + 2.16i)5-s + (2.37 + 1.17i)7-s + (−0.980 + 2.03i)8-s + (−3.49 + 5.12i)10-s + (−0.655 + 4.34i)11-s + (−3.43 − 2.74i)13-s + (4.24 + 0.363i)14-s + (−0.361 + 4.82i)16-s + (3.21 − 0.992i)17-s + (−0.776 − 0.448i)19-s + (−0.511 + 2.24i)20-s + (1.57 + 6.90i)22-s + (−0.732 + 2.37i)23-s + ⋯
L(s)  = 1  + (1.06 − 0.416i)2-s + (0.218 − 0.203i)4-s + (−1.42 + 0.969i)5-s + (0.896 + 0.443i)7-s + (−0.346 + 0.720i)8-s + (−1.10 + 1.62i)10-s + (−0.197 + 1.31i)11-s + (−0.953 − 0.760i)13-s + (1.13 + 0.0972i)14-s + (−0.0903 + 1.20i)16-s + (0.780 − 0.240i)17-s + (−0.178 − 0.102i)19-s + (−0.114 + 0.501i)20-s + (0.336 + 1.47i)22-s + (−0.152 + 0.494i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.239 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.239 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.239 - 0.971i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (278, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.239 - 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29543 + 1.01522i\)
\(L(\frac12)\) \(\approx\) \(1.29543 + 1.01522i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.37 - 1.17i)T \)
good2 \( 1 + (-1.50 + 0.588i)T + (1.46 - 1.36i)T^{2} \)
5 \( 1 + (3.18 - 2.16i)T + (1.82 - 4.65i)T^{2} \)
11 \( 1 + (0.655 - 4.34i)T + (-10.5 - 3.24i)T^{2} \)
13 \( 1 + (3.43 + 2.74i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (-3.21 + 0.992i)T + (14.0 - 9.57i)T^{2} \)
19 \( 1 + (0.776 + 0.448i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.732 - 2.37i)T + (-19.0 - 12.9i)T^{2} \)
29 \( 1 + (-8.30 - 1.89i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 + (-4.67 + 2.69i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.45 - 5.06i)T + (2.76 + 36.8i)T^{2} \)
41 \( 1 + (5.36 + 2.58i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (2.54 - 1.22i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-1.55 - 3.96i)T + (-34.4 + 31.9i)T^{2} \)
53 \( 1 + (8.87 + 9.56i)T + (-3.96 + 52.8i)T^{2} \)
59 \( 1 + (6.88 + 4.69i)T + (21.5 + 54.9i)T^{2} \)
61 \( 1 + (-2.93 + 3.16i)T + (-4.55 - 60.8i)T^{2} \)
67 \( 1 + (1.42 + 2.47i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.30 + 1.66i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (-3.88 - 1.52i)T + (53.5 + 49.6i)T^{2} \)
79 \( 1 + (1.80 - 3.13i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.34 - 10.4i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (3.95 - 0.596i)T + (85.0 - 26.2i)T^{2} \)
97 \( 1 - 18.1iT - 97T^{2} \)
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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

−11.66807583417981968786202190153, −10.76423283944870666013118001815, −9.797577612802556039967369420796, −8.073911635499423030574193019948, −7.85447699844047461381232011919, −6.63394743473228841639506698682, −5.05752568705339224910185911057, −4.56495107240327441967560016302, −3.32755174798230717932394957682, −2.44731730642310555427722273029, 0.77346800173218867094660964790, 3.30614709663208077857241777360, 4.45779719628557913354219840472, 4.75030235643802785458452282850, 5.97366651755077988109441585885, 7.23789840479000193314210045599, 8.109557190535243890210151642694, 8.769959909355701905043037376855, 10.13761240151840164347137895736, 11.32592880539812154930711562713

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