Properties

Label 2-21e2-7.4-c3-0-15
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  + (2.02 + 3.51i)2-s + (−4.22 + 7.31i)4-s + (−4.96 − 8.59i)5-s − 1.80·8-s + (20.1 − 34.8i)10-s + (6.76 − 11.7i)11-s − 18.5·13-s + (30.1 + 52.1i)16-s + (−46.8 + 81.1i)17-s + (65.9 + 114. i)19-s + 83.7·20-s + 54.9·22-s + (99.1 + 171. i)23-s + (13.2 − 23.0i)25-s + (−37.6 − 65.1i)26-s + ⋯
L(s)  = 1  + (0.716 + 1.24i)2-s + (−0.527 + 0.914i)4-s + (−0.443 − 0.768i)5-s − 0.0799·8-s + (0.636 − 1.10i)10-s + (0.185 − 0.321i)11-s − 0.395·13-s + (0.470 + 0.815i)16-s + (−0.668 + 1.15i)17-s + (0.796 + 1.37i)19-s + 0.936·20-s + 0.532·22-s + (0.898 + 1.55i)23-s + (0.106 − 0.184i)25-s + (−0.283 − 0.491i)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

\(L(2)\) \(\approx\) \(2.514934639\)
\(L(\frac12)\) \(\approx\) \(2.514934639\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.02 - 3.51i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (4.96 + 8.59i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-6.76 + 11.7i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 18.5T + 2.19e3T^{2} \)
17 \( 1 + (46.8 - 81.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-65.9 - 114. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-99.1 - 171. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 188.T + 2.43e4T^{2} \)
31 \( 1 + (41.9 - 72.6i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (40.0 + 69.4i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 385.T + 6.89e4T^{2} \)
43 \( 1 + 397.T + 7.95e4T^{2} \)
47 \( 1 + (-136. - 235. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-18.4 + 32.0i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-197. + 342. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-6.73 - 11.6i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (170. - 294. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 211.T + 3.57e5T^{2} \)
73 \( 1 + (-243. + 420. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (146. + 254. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 889.T + 5.71e5T^{2} \)
89 \( 1 + (-572. - 991. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.38e3T + 9.12e5T^{2} \)
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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

−11.14739203918921959386516933347, −10.03315627741140221730448829977, −8.809602463791677749827847897369, −8.081792659978950372358360475554, −7.28019200469454025052924839880, −6.21168031826335158565966914180, −5.38089658401553641811882475818, −4.47377542469683242664378105796, −3.51936430320641717766519310541, −1.35965778053180748168890093209, 0.70813922531276922751971213625, 2.50415651788798355253027539806, 3.03088797466165391203133167119, 4.39480154278735111753150064644, 5.04998781110471260960950220512, 6.78986779720078252098851610330, 7.34427386020763096623207139832, 8.843771023221301745730219729501, 9.829691100565955159256975755586, 10.72826517901621522487432201665

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