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Properties

Label 2-21e2-63.25-c1-0-13
Degree $2$
Conductor $441$
Sign $0.445 - 0.895i$
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  − 2.71·2-s + (1.16 + 1.27i)3-s + 5.37·4-s + (0.793 − 1.37i)5-s + (−3.17 − 3.46i)6-s − 9.15·8-s + (−0.264 + 2.98i)9-s + (−2.15 + 3.73i)10-s + (0.674 + 1.16i)11-s + (6.28 + 6.86i)12-s + (1.58 + 2.75i)13-s + (2.68 − 0.593i)15-s + 14.1·16-s + (−1.40 + 2.42i)17-s + (0.717 − 8.11i)18-s + (−0.312 − 0.541i)19-s + ⋯
L(s)  = 1  − 1.91·2-s + (0.675 + 0.737i)3-s + 2.68·4-s + (0.354 − 0.614i)5-s + (−1.29 − 1.41i)6-s − 3.23·8-s + (−0.0880 + 0.996i)9-s + (−0.681 + 1.17i)10-s + (0.203 + 0.352i)11-s + (1.81 + 1.98i)12-s + (0.440 + 0.763i)13-s + (0.692 − 0.153i)15-s + 3.52·16-s + (−0.339 + 0.588i)17-s + (0.169 − 1.91i)18-s + (−0.0717 − 0.124i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.445 - 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.445 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.445 - 0.895i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (214, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.445 - 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.674664 + 0.417694i\)
\(L(\frac12)\) \(\approx\) \(0.674664 + 0.417694i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.16 - 1.27i)T \)
7 \( 1 \)
good2 \( 1 + 2.71T + 2T^{2} \)
5 \( 1 + (-0.793 + 1.37i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.674 - 1.16i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.58 - 2.75i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.40 - 2.42i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.312 + 0.541i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.142 + 0.246i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.27 + 3.93i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.43T + 31T^{2} \)
37 \( 1 + (4.01 + 6.94i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.01 - 8.68i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.12 - 5.42i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 11.1T + 47T^{2} \)
53 \( 1 + (1.39 - 2.41i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 4.57T + 59T^{2} \)
61 \( 1 + 0.385T + 61T^{2} \)
67 \( 1 + 2.53T + 67T^{2} \)
71 \( 1 + 1.45T + 71T^{2} \)
73 \( 1 + (-0.234 + 0.405i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 15.7T + 79T^{2} \)
83 \( 1 + (6.99 - 12.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.29 - 2.24i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.22 + 12.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.87746002611945963641421793347, −10.02838439065942138287200466910, −9.373400004094033637981176800399, −8.754234201526120759916716026280, −8.111404210125647670918748711615, −7.03543117575711018198561435532, −5.93019334087303924373227787774, −4.31071129741481364638034132726, −2.67350675646958435428842309023, −1.49816103082866179627002666543, 0.928234554131100956164777167119, 2.33542686076134170865572056051, 3.19534711240030223733064881043, 5.96451514207410626838758353638, 6.76792916037346448485280288267, 7.44185409525636682761894399478, 8.491402195884200391944471638713, 8.859450964393601227251167320600, 10.02555924155856266152723193104, 10.56386928105260000459600734941

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