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Properties

Label 2-21e2-63.4-c1-0-4
Degree $2$
Conductor $441$
Sign $-0.976 + 0.215i$
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  + (−0.335 + 0.580i)2-s + (0.377 + 1.69i)3-s + (0.775 + 1.34i)4-s − 1.42·5-s + (−1.10 − 0.347i)6-s − 2.38·8-s + (−2.71 + 1.27i)9-s + (0.477 − 0.827i)10-s − 4.93·11-s + (−1.97 + 1.81i)12-s + (1.37 − 2.38i)13-s + (−0.537 − 2.40i)15-s + (−0.752 + 1.30i)16-s + (−0.559 + 0.969i)17-s + (0.169 − 2.00i)18-s + (2.00 + 3.47i)19-s + ⋯
L(s)  = 1  + (−0.236 + 0.410i)2-s + (0.217 + 0.975i)3-s + (0.387 + 0.671i)4-s − 0.637·5-s + (−0.452 − 0.141i)6-s − 0.841·8-s + (−0.905 + 0.425i)9-s + (0.151 − 0.261i)10-s − 1.48·11-s + (−0.570 + 0.524i)12-s + (0.381 − 0.661i)13-s + (−0.138 − 0.621i)15-s + (−0.188 + 0.326i)16-s + (−0.135 + 0.235i)17-s + (0.0399 − 0.472i)18-s + (0.460 + 0.797i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.215i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.976 + 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.976 + 0.215i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.976 + 0.215i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0904560 - 0.830896i\)
\(L(\frac12)\) \(\approx\) \(0.0904560 - 0.830896i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.377 - 1.69i)T \)
7 \( 1 \)
good2 \( 1 + (0.335 - 0.580i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 1.42T + 5T^{2} \)
11 \( 1 + 4.93T + 11T^{2} \)
13 \( 1 + (-1.37 + 2.38i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.559 - 0.969i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.00 - 3.47i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 5.43T + 23T^{2} \)
29 \( 1 + (-3.40 - 5.89i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.25 - 2.17i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.709 - 1.22i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.124 - 0.215i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.498 + 0.863i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.73 - 8.20i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.410 - 0.710i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.29 + 5.70i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.0376 + 0.0651i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.29 - 10.9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.0804T + 71T^{2} \)
73 \( 1 + (5.34 - 9.25i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.922 + 1.59i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.23 - 12.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.76 + 11.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.70 + 4.67i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.35870953163875708745766688527, −10.73080609901646952858132025723, −9.818301931406127807641922802385, −8.566336118755940959242351233858, −8.120228568716174281037796538664, −7.28875372768930470197274128165, −5.88334972867937085871438703833, −4.87385539075788704550751035034, −3.53445161496973953857417050869, −2.83058109638267546641073361954, 0.52162292767903139563300245412, 2.14367843695268331464457863736, 3.11834756838209408148436260680, 4.93580382696645997619207977081, 6.05439776694723459768144076614, 7.01937614039901090335926708246, 7.83193757669430308711289470505, 8.798464577928393646140014059294, 9.763920971781409234358054548946, 10.90161460836315946147768992025

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