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Properties

Label 2-21e2-63.4-c1-0-30
Degree $2$
Conductor $441$
Sign $0.0977 + 0.995i$
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  + (0.849 − 1.47i)2-s + (1.58 − 0.709i)3-s + (−0.444 − 0.769i)4-s + 0.949·5-s + (0.298 − 2.92i)6-s + 1.88·8-s + (1.99 − 2.24i)9-s + (0.806 − 1.39i)10-s − 0.588·11-s + (−1.24 − 0.901i)12-s + (−2.50 + 4.34i)13-s + (1.49 − 0.673i)15-s + (2.49 − 4.31i)16-s + (−3.79 + 6.56i)17-s + (−1.60 − 4.83i)18-s + (−2.23 − 3.86i)19-s + ⋯
L(s)  = 1  + (0.600 − 1.04i)2-s + (0.912 − 0.409i)3-s + (−0.222 − 0.384i)4-s + 0.424·5-s + (0.121 − 1.19i)6-s + 0.667·8-s + (0.664 − 0.747i)9-s + (0.255 − 0.441i)10-s − 0.177·11-s + (−0.360 − 0.260i)12-s + (−0.696 + 1.20i)13-s + (0.387 − 0.173i)15-s + (0.623 − 1.07i)16-s + (−0.919 + 1.59i)17-s + (−0.378 − 1.14i)18-s + (−0.511 − 0.886i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0977 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0977 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.0977 + 0.995i$
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: \((2,\ 441,\ (\ :1/2),\ 0.0977 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.01853 - 1.82996i\)
\(L(\frac12)\) \(\approx\) \(2.01853 - 1.82996i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.58 + 0.709i)T \)
7 \( 1 \)
good2 \( 1 + (-0.849 + 1.47i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 0.949T + 5T^{2} \)
11 \( 1 + 0.588T + 11T^{2} \)
13 \( 1 + (2.50 - 4.34i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.79 - 6.56i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.23 + 3.86i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 2.47T + 23T^{2} \)
29 \( 1 + (2.73 + 4.74i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.03 + 5.26i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.49 - 6.05i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.527 - 0.913i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.49 + 6.05i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.73 - 6.47i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.46 - 5.99i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.21 - 9.03i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.82 + 10.0i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.93 - 10.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.30T + 71T^{2} \)
73 \( 1 + (-2.23 + 3.86i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.666 + 1.15i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.84 + 4.92i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.421 - 0.730i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.70 + 2.94i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.12546758322230721740854759671, −10.03768191556233731345974223469, −9.294903130375048229669260866594, −8.267982591217058281788478215667, −7.23697992462560131660790939150, −6.25819592063978839046956242423, −4.57562024202278651359949973422, −3.84255700542230919069724662701, −2.44397601713889587336034743851, −1.84477717159881129172498998362, 2.16646548128725599645950592915, 3.52499583555760908786661765279, 4.87038562539922401467346898322, 5.41784634316967065162313118748, 6.78818355436085575827277022946, 7.54386235537261510101435394317, 8.382542418839602380473443652867, 9.491893794139158046381649322329, 10.23665801280343982253206802046, 11.14142822624886713899821781652

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