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Properties

Label 2-21e2-147.62-c1-0-11
Degree $2$
Conductor $441$
Sign $-0.287 + 0.957i$
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  + (−0.843 + 0.192i)2-s + (−1.12 + 0.542i)4-s + (−0.0384 − 0.0482i)5-s + (−2.43 + 1.03i)7-s + (2.19 − 1.75i)8-s + (0.0417 + 0.0333i)10-s + (3.40 − 0.776i)11-s + (−4.60 + 1.05i)13-s + (1.85 − 1.34i)14-s + (0.0416 − 0.0522i)16-s + (−2.52 − 1.21i)17-s − 3.97i·19-s + (0.0695 + 0.0335i)20-s + (−2.71 + 1.30i)22-s + (−2.91 − 6.04i)23-s + ⋯
L(s)  = 1  + (−0.596 + 0.136i)2-s + (−0.563 + 0.271i)4-s + (−0.0172 − 0.0215i)5-s + (−0.920 + 0.391i)7-s + (0.777 − 0.620i)8-s + (0.0132 + 0.0105i)10-s + (1.02 − 0.234i)11-s + (−1.27 + 0.291i)13-s + (0.495 − 0.358i)14-s + (0.0104 − 0.0130i)16-s + (−0.611 − 0.294i)17-s − 0.912i·19-s + (0.0155 + 0.00749i)20-s + (−0.579 + 0.279i)22-s + (−0.606 − 1.26i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.287 + 0.957i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.287 + 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.209865 - 0.282058i\)
\(L(\frac12)\) \(\approx\) \(0.209865 - 0.282058i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (2.43 - 1.03i)T \)
good2 \( 1 + (0.843 - 0.192i)T + (1.80 - 0.867i)T^{2} \)
5 \( 1 + (0.0384 + 0.0482i)T + (-1.11 + 4.87i)T^{2} \)
11 \( 1 + (-3.40 + 0.776i)T + (9.91 - 4.77i)T^{2} \)
13 \( 1 + (4.60 - 1.05i)T + (11.7 - 5.64i)T^{2} \)
17 \( 1 + (2.52 + 1.21i)T + (10.5 + 13.2i)T^{2} \)
19 \( 1 + 3.97iT - 19T^{2} \)
23 \( 1 + (2.91 + 6.04i)T + (-14.3 + 17.9i)T^{2} \)
29 \( 1 + (-2.79 + 5.79i)T + (-18.0 - 22.6i)T^{2} \)
31 \( 1 - 5.72iT - 31T^{2} \)
37 \( 1 + (7.25 + 3.49i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + (0.544 + 0.683i)T + (-9.12 + 39.9i)T^{2} \)
43 \( 1 + (0.942 - 1.18i)T + (-9.56 - 41.9i)T^{2} \)
47 \( 1 + (2.98 + 13.0i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (-2.89 - 6.01i)T + (-33.0 + 41.4i)T^{2} \)
59 \( 1 + (4.09 - 5.13i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (-1.01 + 2.11i)T + (-38.0 - 47.6i)T^{2} \)
67 \( 1 + 6.92T + 67T^{2} \)
71 \( 1 + (1.87 + 3.89i)T + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (-12.7 - 2.90i)T + (65.7 + 31.6i)T^{2} \)
79 \( 1 + 6.76T + 79T^{2} \)
83 \( 1 + (0.469 - 2.05i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (2.55 - 11.1i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 - 19.0iT - 97T^{2} \)
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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

−10.53133019464172058139231818268, −9.764790054904510704374331370813, −9.016950277134469620742959421935, −8.430818041159975359262235977688, −7.05514208304899054585670992916, −6.51250407308343860669677500168, −4.92694907240687157262403749165, −3.98233210459784212663921996934, −2.53455193961024625370366708712, −0.27662012669444101325749702979, 1.59476403975978257798935492387, 3.44280947874479427093398739854, 4.54655801708255826951218259614, 5.73167880006403662995071141425, 6.90872021046075254965677470370, 7.78429473759380893563173022298, 8.943191288181241292745025255466, 9.684531804124711704796965429982, 10.12961800922070796565211960029, 11.21216863903548964369157718131

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