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Properties

Label 2-21e2-21.5-c3-0-14
Degree $2$
Conductor $441$
Sign $-0.851 - 0.524i$
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  + (3.91 + 2.26i)2-s + (6.22 + 10.7i)4-s + (−0.632 + 1.09i)5-s + 20.1i·8-s + (−4.95 + 2.86i)10-s + (−36.0 + 20.7i)11-s + 85.7i·13-s + (4.27 − 7.39i)16-s + (38.8 + 67.3i)17-s + (−42.1 − 24.3i)19-s − 15.7·20-s − 188.·22-s + (−78.7 − 45.4i)23-s + (61.6 + 106. i)25-s + (−193. + 335. i)26-s + ⋯
L(s)  = 1  + (1.38 + 0.799i)2-s + (0.778 + 1.34i)4-s + (−0.0566 + 0.0980i)5-s + 0.890i·8-s + (−0.156 + 0.0905i)10-s + (−0.987 + 0.570i)11-s + 1.82i·13-s + (0.0667 − 0.115i)16-s + (0.554 + 0.961i)17-s + (−0.509 − 0.293i)19-s − 0.176·20-s − 1.82·22-s + (−0.714 − 0.412i)23-s + (0.493 + 0.854i)25-s + (−1.46 + 2.53i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.851 - 0.524i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.851 - 0.524i$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ -0.851 - 0.524i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.322807956\)
\(L(\frac12)\) \(\approx\) \(3.322807956\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.91 - 2.26i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + (0.632 - 1.09i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (36.0 - 20.7i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 85.7iT - 2.19e3T^{2} \)
17 \( 1 + (-38.8 - 67.3i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (42.1 + 24.3i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (78.7 + 45.4i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 151. iT - 2.43e4T^{2} \)
31 \( 1 + (76.3 - 44.0i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (45.2 - 78.4i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 383.T + 6.89e4T^{2} \)
43 \( 1 + 227.T + 7.95e4T^{2} \)
47 \( 1 + (-69.5 + 120. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-289. + 167. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (440. + 762. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (11.3 + 6.57i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-221. - 383. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 341. iT - 3.57e5T^{2} \)
73 \( 1 + (-798. + 460. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-206. + 357. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 954.T + 5.71e5T^{2} \)
89 \( 1 + (14.8 - 25.7i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.19e3iT - 9.12e5T^{2} \)
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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

−11.31997688136249425178172492034, −10.31063928007496895076631794005, −9.145642741636010682100749344206, −7.970867054343313626861553870560, −7.04059609683376970693750912955, −6.37016437229478487623205921378, −5.25560221639394711065237212354, −4.45160912028602272276054173587, −3.48912034307449561641473926674, −2.02601964695031155260099189148, 0.64531099126977892958229515523, 2.46307608294943041055149095904, 3.19698849277697181239204714591, 4.36660304002845042097832903215, 5.48556268532310529432615463595, 5.89789347926576689517193771391, 7.59757154703450214567650282437, 8.362892145488303467884943409452, 9.937870468546768984424822393090, 10.58216697413613777815283979506

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