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Properties

Label 2-21e2-63.5-c1-0-13
Degree $2$
Conductor $441$
Sign $0.807 - 0.590i$
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  − 1.17i·2-s + (0.278 + 1.70i)3-s + 0.611·4-s + (2.16 + 3.75i)5-s + (2.01 − 0.327i)6-s − 3.07i·8-s + (−2.84 + 0.950i)9-s + (4.42 − 2.55i)10-s + (1.87 + 1.08i)11-s + (0.170 + 1.04i)12-s + (−2.25 − 1.30i)13-s + (−5.81 + 4.74i)15-s − 2.40·16-s + (−0.585 − 1.01i)17-s + (1.12 + 3.35i)18-s + (−2.09 − 1.20i)19-s + ⋯
L(s)  = 1  − 0.833i·2-s + (0.160 + 0.987i)3-s + 0.305·4-s + (0.968 + 1.67i)5-s + (0.822 − 0.133i)6-s − 1.08i·8-s + (−0.948 + 0.316i)9-s + (1.39 − 0.807i)10-s + (0.564 + 0.325i)11-s + (0.0491 + 0.301i)12-s + (−0.624 − 0.360i)13-s + (−1.50 + 1.22i)15-s − 0.600·16-s + (−0.142 − 0.245i)17-s + (0.264 + 0.790i)18-s + (−0.480 − 0.277i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.807 - 0.590i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.807 - 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.807 - 0.590i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.807 - 0.590i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.78546 + 0.583341i\)
\(L(\frac12)\) \(\approx\) \(1.78546 + 0.583341i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.278 - 1.70i)T \)
7 \( 1 \)
good2 \( 1 + 1.17iT - 2T^{2} \)
5 \( 1 + (-2.16 - 3.75i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.87 - 1.08i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.25 + 1.30i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.585 + 1.01i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.09 + 1.20i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.16 + 1.82i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.589 + 0.340i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.55iT - 31T^{2} \)
37 \( 1 + (-2.55 + 4.42i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.68 + 6.38i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.12 + 3.68i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 7.14T + 47T^{2} \)
53 \( 1 + (-2.79 + 1.61i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 5.83T + 59T^{2} \)
61 \( 1 + 7.17iT - 61T^{2} \)
67 \( 1 - 6.65T + 67T^{2} \)
71 \( 1 - 1.95iT - 71T^{2} \)
73 \( 1 + (10.3 - 5.95i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 9.75T + 79T^{2} \)
83 \( 1 + (0.796 + 1.37i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.04 - 5.28i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.36 + 1.36i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.89856453139512749646552429827, −10.44523813303409843141215878942, −9.813921922639791678886241724327, −9.012505345395538811389669583925, −7.26065232937983948266742829786, −6.60021196429691847855759648699, −5.47474794389692522789647157985, −3.94480211510182871370900976317, −2.90233861365798455443191382977, −2.24499915206734731655205595052, 1.30364465242976257138419935697, 2.39556548198577842626253554432, 4.58606899642738183175652055383, 5.72380684833728591468364528134, 6.21075664153240348440718075456, 7.30486480229134592940526782981, 8.302095838612420837203758930397, 8.860296791354841309056495986605, 9.772830809764183083874337501414, 11.34871499657314862984092044911

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