Properties

Label 2-21e2-7.4-c3-0-37
Degree $2$
Conductor $441$
Sign $0.605 + 0.795i$
Analytic cond. $26.0198$
Root an. cond. $5.10096$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + 1.73i)2-s + (2.00 − 3.46i)4-s + (−3.5 − 6.06i)5-s + 24·8-s + (7 − 12.1i)10-s + (−2.5 + 4.33i)11-s + 14·13-s + (8.00 + 13.8i)16-s + (10.5 − 18.1i)17-s + (24.5 + 42.4i)19-s − 28·20-s − 10·22-s + (−79.5 − 137. i)23-s + (38 − 65.8i)25-s + (14 + 24.2i)26-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.250 − 0.433i)4-s + (−0.313 − 0.542i)5-s + 1.06·8-s + (0.221 − 0.383i)10-s + (−0.0685 + 0.118i)11-s + 0.298·13-s + (0.125 + 0.216i)16-s + (0.149 − 0.259i)17-s + (0.295 + 0.512i)19-s − 0.313·20-s − 0.0969·22-s + (−0.720 − 1.24i)23-s + (0.303 − 0.526i)25-s + (0.105 + 0.182i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.300631942\)
\(L(\frac12)\) \(\approx\) \(2.300631942\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good2 \( 1 + (-1 - 1.73i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (3.5 + 6.06i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (2.5 - 4.33i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 14T + 2.19e3T^{2} \)
17 \( 1 + (-10.5 + 18.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-24.5 - 42.4i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (79.5 + 137. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 58T + 2.43e4T^{2} \)
31 \( 1 + (-73.5 + 127. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (109.5 + 189. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 350T + 6.89e4T^{2} \)
43 \( 1 + 124T + 7.95e4T^{2} \)
47 \( 1 + (262.5 + 454. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-151.5 + 262. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-52.5 + 90.9i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (206.5 + 357. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (207.5 - 359. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 432T + 3.57e5T^{2} \)
73 \( 1 + (556.5 - 963. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-51.5 - 89.2i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 1.09e3T + 5.71e5T^{2} \)
89 \( 1 + (-164.5 - 284. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 882T + 9.12e5T^{2} \)
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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

−10.52422566606867945048235042321, −9.757853118851677833741115583815, −8.536604207943450782541083358859, −7.74575439156514383218661395330, −6.71947185453467222733909125214, −5.83583982496086992867010804308, −4.89261457818669842278298096500, −3.94303951354760821856082956948, −2.16309725505555033018869092385, −0.68262892341191589181056911257, 1.49809922045833177963939657630, 2.91462460474436470892873893938, 3.63497980941682298939895597841, 4.80494699966361850064616208640, 6.16104310389652089570864482625, 7.28735193023372115126915209954, 7.898108879276438483713970952290, 9.095710999589291826325230425474, 10.25196773183186935003118302094, 11.03926514914109270674773030661

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