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Properties

Label 2-21e2-7.4-c3-0-1
Degree $2$
Conductor $441$
Sign $0.991 - 0.126i$
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  + (−1.5 − 2.59i)2-s + (−0.5 + 0.866i)4-s + (−9 − 15.5i)5-s − 21·8-s + (−27 + 46.7i)10-s + (−18 + 31.1i)11-s − 34·13-s + (35.5 + 61.4i)16-s + (21 − 36.3i)17-s + (62 + 107. i)19-s + 18.0·20-s + 108·22-s + (−99.5 + 172. i)25-s + (51 + 88.3i)26-s − 102·29-s + ⋯
L(s)  = 1  + (−0.530 − 0.918i)2-s + (−0.0625 + 0.108i)4-s + (−0.804 − 1.39i)5-s − 0.928·8-s + (−0.853 + 1.47i)10-s + (−0.493 + 0.854i)11-s − 0.725·13-s + (0.554 + 0.960i)16-s + (0.299 − 0.518i)17-s + (0.748 + 1.29i)19-s + 0.201·20-s + 1.04·22-s + (−0.796 + 1.37i)25-s + (0.384 + 0.666i)26-s − 0.653·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.991 - 0.126i$
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: \((2,\ 441,\ (\ :3/2),\ 0.991 - 0.126i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3620128244\)
\(L(\frac12)\) \(\approx\) \(0.3620128244\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.5 + 2.59i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (9 + 15.5i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (18 - 31.1i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 34T + 2.19e3T^{2} \)
17 \( 1 + (-21 + 36.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-62 - 107. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 102T + 2.43e4T^{2} \)
31 \( 1 + (-80 + 138. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (199 + 344. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 318T + 6.89e4T^{2} \)
43 \( 1 + 268T + 7.95e4T^{2} \)
47 \( 1 + (-120 - 207. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (249 - 431. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (66 - 114. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (199 + 344. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (46 - 79.6i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 720T + 3.57e5T^{2} \)
73 \( 1 + (-251 + 434. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-512 - 886. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 204T + 5.71e5T^{2} \)
89 \( 1 + (-177 - 306. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 286T + 9.12e5T^{2} \)
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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

−10.72244269458304830184612150227, −9.652805353293206174294071375458, −9.295288168287308177185461450481, −8.080118917161985768171813237884, −7.44750598350123986931806370584, −5.73381812900763528845855673376, −4.82719764011178492039448634830, −3.66704798221737614130064441882, −2.18558658586899059421386427078, −0.971526907687978120401894388021, 0.17148708271327106614573697382, 2.78478356469861248565595900626, 3.44563953535668772175388960122, 5.17805637587241581848038960416, 6.41835466325026410003292129222, 7.06680650051527840579043149330, 7.78902893496095748586125694779, 8.532309098398154091855574850792, 9.697469467794690324601294146015, 10.69725525511903010211118579835

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