Properties

Label 2-21e2-63.25-c1-0-11
Degree $2$
Conductor $441$
Sign $0.386 - 0.922i$
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  − 0.879·2-s + (1.70 + 0.300i)3-s − 1.22·4-s + (−0.673 + 1.16i)5-s + (−1.49 − 0.264i)6-s + 2.83·8-s + (2.81 + 1.02i)9-s + (0.592 − 1.02i)10-s + (−0.826 − 1.43i)11-s + (−2.09 − 0.368i)12-s + (1.68 + 2.91i)13-s + (−1.49 + 1.78i)15-s − 0.0418·16-s + (−0.233 + 0.405i)17-s + (−2.47 − 0.902i)18-s + (1.61 + 2.79i)19-s + ⋯
L(s)  = 1  − 0.621·2-s + (0.984 + 0.173i)3-s − 0.613·4-s + (−0.301 + 0.521i)5-s + (−0.612 − 0.107i)6-s + 1.00·8-s + (0.939 + 0.342i)9-s + (0.187 − 0.324i)10-s + (−0.249 − 0.431i)11-s + (−0.604 − 0.106i)12-s + (0.467 + 0.809i)13-s + (−0.387 + 0.461i)15-s − 0.0104·16-s + (−0.0567 + 0.0982i)17-s + (−0.584 − 0.212i)18-s + (0.370 + 0.641i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.386 - 0.922i$
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: \((2,\ 441,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.938281 + 0.624127i\)
\(L(\frac12)\) \(\approx\) \(0.938281 + 0.624127i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.70 - 0.300i)T \)
7 \( 1 \)
good2 \( 1 + 0.879T + 2T^{2} \)
5 \( 1 + (0.673 - 1.16i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.826 + 1.43i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.68 - 2.91i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.233 - 0.405i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.61 - 2.79i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.47 - 7.74i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.13 - 5.42i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 9.23T + 31T^{2} \)
37 \( 1 + (4.61 + 7.99i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.70 + 2.95i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.20 + 3.82i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 9.35T + 47T^{2} \)
53 \( 1 + (-0.286 + 0.497i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 10.3T + 59T^{2} \)
61 \( 1 - 7.63T + 61T^{2} \)
67 \( 1 - 0.596T + 67T^{2} \)
71 \( 1 + 0.554T + 71T^{2} \)
73 \( 1 + (1.02 - 1.77i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 2.40T + 79T^{2} \)
83 \( 1 + (-7.52 + 13.0i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.54 + 7.86i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.949 + 1.64i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.94773737370989377465693898109, −10.21549274299031568212645884775, −9.312895755932199549681463014363, −8.706430093793724502473165467113, −7.78427391624440267848488068139, −7.12873892435727726138795066645, −5.52996336151396633458072319392, −4.12621263038470736962608168780, −3.38157216903191666872861020199, −1.66855144921805087931143060598, 0.882887523815293774766513605622, 2.61051943714259769912099645038, 4.10203968912438284950744404112, 4.85554584725551785747773197947, 6.51555933910185661041971201222, 7.84429591618065059960494517379, 8.219413747537252723620505317214, 8.983042859746877906005652345879, 9.925309753486600461271670271523, 10.49755563732199483507551508321

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