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Properties

Label 2-21e2-21.5-c3-0-10
Degree $2$
Conductor $441$
Sign $0.652 - 0.757i$
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  + (−3.16 − 1.82i)2-s + (2.67 + 4.63i)4-s + (−8.69 + 15.0i)5-s + 9.68i·8-s + (55.0 − 31.7i)10-s + (28.2 − 16.2i)11-s − 42.4i·13-s + (39.0 − 67.7i)16-s + (−42.8 − 74.1i)17-s + (−59.9 − 34.5i)19-s − 93.0·20-s − 118.·22-s + (144. + 83.5i)23-s + (−88.8 − 153. i)25-s + (−77.5 + 134. i)26-s + ⋯
L(s)  = 1  + (−1.11 − 0.645i)2-s + (0.334 + 0.578i)4-s + (−0.778 + 1.34i)5-s + 0.428i·8-s + (1.74 − 1.00i)10-s + (0.773 − 0.446i)11-s − 0.906i·13-s + (0.610 − 1.05i)16-s + (−0.610 − 1.05i)17-s + (−0.723 − 0.417i)19-s − 1.04·20-s − 1.15·22-s + (1.31 + 0.757i)23-s + (−0.710 − 1.23i)25-s + (−0.585 + 1.01i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.652 - 0.757i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.652 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.652 - 0.757i$
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: \((2,\ 441,\ (\ :3/2),\ 0.652 - 0.757i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6248001892\)
\(L(\frac12)\) \(\approx\) \(0.6248001892\)
\(L(\frac{5}{2})\) 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 \( 1 + (3.16 + 1.82i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + (8.69 - 15.0i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-28.2 + 16.2i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 42.4iT - 2.19e3T^{2} \)
17 \( 1 + (42.8 + 74.1i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (59.9 + 34.5i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-144. - 83.5i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 254. iT - 2.43e4T^{2} \)
31 \( 1 + (-281. + 162. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (172. - 299. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 182.T + 6.89e4T^{2} \)
43 \( 1 - 140.T + 7.95e4T^{2} \)
47 \( 1 + (-21.5 + 37.2i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (167. - 96.5i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (55.6 + 96.3i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (78.4 + 45.2i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-524. - 908. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 464. iT - 3.57e5T^{2} \)
73 \( 1 + (-158. + 91.3i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-119. + 206. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 375.T + 5.71e5T^{2} \)
89 \( 1 + (719. - 1.24e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 638. iT - 9.12e5T^{2} \)
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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

−10.87794403943905262150312084226, −10.07356200229292846992713842268, −9.077686361128724015134504287262, −8.282810278561988708715240354968, −7.28487861972442567592277788364, −6.54277347678404402711412587797, −4.98144382392144069864270665817, −3.36353346139255964759914837464, −2.65104974176067303132976361742, −0.915965753341539143452191635751, 0.41403083410946347706482916988, 1.62542081763773668114499772111, 3.99327713486657558030482900486, 4.58808401730765783261779825878, 6.25198213166972677959242977204, 7.02585793369485822878016707300, 8.178220491904165591876591623392, 8.662300358209027029048301752715, 9.260303461644446580405208278185, 10.28846751623367470467707336764

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