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Properties

Label 2-21e2-7.4-c3-0-40
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  + (−0.5 − 0.866i)2-s + (3.5 − 6.06i)4-s + (6 + 10.3i)5-s − 15·8-s + (6 − 10.3i)10-s + (10 − 17.3i)11-s − 84·13-s + (−20.5 − 35.5i)16-s + (−48 + 83.1i)17-s + (−6 − 10.3i)19-s + 84·20-s − 20·22-s + (−88 − 152. i)23-s + (−9.5 + 16.4i)25-s + (42 + 72.7i)26-s + ⋯
L(s)  = 1  + (−0.176 − 0.306i)2-s + (0.437 − 0.757i)4-s + (0.536 + 0.929i)5-s − 0.662·8-s + (0.189 − 0.328i)10-s + (0.274 − 0.474i)11-s − 1.79·13-s + (−0.320 − 0.554i)16-s + (−0.684 + 1.18i)17-s + (−0.0724 − 0.125i)19-s + 0.939·20-s − 0.193·22-s + (−0.797 − 1.38i)23-s + (−0.0759 + 0.131i)25-s + (0.316 + 0.548i)26-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

\(L(2)\) \(\approx\) \(0.6320383697\)
\(L(\frac12)\) \(\approx\) \(0.6320383697\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.5 + 0.866i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (-6 - 10.3i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-10 + 17.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 84T + 2.19e3T^{2} \)
17 \( 1 + (48 - 83.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (6 + 10.3i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (88 + 152. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 58T + 2.43e4T^{2} \)
31 \( 1 + (-132 + 228. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (129 + 223. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 6.89e4T^{2} \)
43 \( 1 - 156T + 7.95e4T^{2} \)
47 \( 1 + (204 + 353. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (361 - 625. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-246 + 426. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-246 - 426. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (206 - 356. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 296T + 3.57e5T^{2} \)
73 \( 1 + (120 - 207. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (388 + 672. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 924T + 5.71e5T^{2} \)
89 \( 1 + (372 + 644. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 168T + 9.12e5T^{2} \)
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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

−10.28174854617387796243136856437, −9.737043166239189948769570730732, −8.616780781641489625535750321950, −7.28201162734646247342150320291, −6.42632031487126227601507658702, −5.76518169382761903071283656267, −4.35941924716965376605716711676, −2.68872160782012637990717871408, −2.04307095388213646035157986692, −0.19031750663660969928956840750, 1.78186458882518267626696745818, 2.97299012254096249770522889307, 4.53500634485210982063676512619, 5.34914329155285171808153018627, 6.72765325243340613813083870491, 7.41120071744287469982313205224, 8.377556848003850117046187218506, 9.392717059801416691943058122991, 9.838476181039644187027877814368, 11.38766796325063979740754421038

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