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Properties

Label 2-21e2-7.4-c3-0-5
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  + (−2.27 − 3.94i)2-s + (−6.38 + 11.0i)4-s + (−8.93 − 15.4i)5-s + 21.6·8-s + (−40.7 + 70.5i)10-s + (−5.69 + 9.86i)11-s + 13.0·13-s + (1.62 + 2.81i)16-s + (−26.6 + 46.1i)17-s + (−21.2 − 36.7i)19-s + 228.·20-s + 51.9·22-s + (76.0 + 131. i)23-s + (−97.2 + 168. i)25-s + (−29.8 − 51.6i)26-s + ⋯
L(s)  = 1  + (−0.805 − 1.39i)2-s + (−0.797 + 1.38i)4-s + (−0.799 − 1.38i)5-s + 0.958·8-s + (−1.28 + 2.23i)10-s + (−0.156 + 0.270i)11-s + 0.279·13-s + (0.0254 + 0.0440i)16-s + (−0.379 + 0.658i)17-s + (−0.256 − 0.443i)19-s + 2.54·20-s + 0.503·22-s + (0.689 + 1.19i)23-s + (−0.777 + 1.34i)25-s + (−0.225 − 0.389i)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

\(L(2)\) \(\approx\) \(0.5778092402\)
\(L(\frac12)\) \(\approx\) \(0.5778092402\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.27 + 3.94i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (8.93 + 15.4i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (5.69 - 9.86i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 13.0T + 2.19e3T^{2} \)
17 \( 1 + (26.6 - 46.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (21.2 + 36.7i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-76.0 - 131. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 186.T + 2.43e4T^{2} \)
31 \( 1 + (78.9 - 136. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (1.87 + 3.24i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 39.3T + 6.89e4T^{2} \)
43 \( 1 - 429.T + 7.95e4T^{2} \)
47 \( 1 + (10.5 + 18.3i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-182. + 316. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-113. + 196. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-325. - 564. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (72.7 - 125. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 368.T + 3.57e5T^{2} \)
73 \( 1 + (-304. + 527. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (455. + 788. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 327.T + 5.71e5T^{2} \)
89 \( 1 + (-18.8 - 32.5i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 722.T + 9.12e5T^{2} \)
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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

−10.74917611004137304287377600655, −9.565432580772252170377594611831, −8.932864231112116151034091458713, −8.308282601278761784273588271041, −7.31204604046330207149053971898, −5.52509322311381321164372341732, −4.33369932422485648256194479638, −3.47108189991406044327049481374, −1.89628411731295137654806048494, −0.839381615845157463671449478460, 0.35321346059477544382161028521, 2.71524187246411102586192554592, 4.04584860930676832018346688696, 5.56360517462509607663636028792, 6.52815608767002755487985508318, 7.18377890094081986308921365547, 7.87146384319089151740365085758, 8.760028514639979396517592506675, 9.726457820584426903146914672645, 10.78140937321747093945322771594

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