Properties

Label 2-21e2-7.4-c3-0-29
Degree $2$
Conductor $441$
Sign $0.827 - 0.561i$
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  + (0.979 + 1.69i)2-s + (2.07 − 3.60i)4-s + (5.94 + 10.2i)5-s + 23.8·8-s + (−11.6 + 20.1i)10-s + (18.2 − 31.5i)11-s + 0.964·13-s + (6.71 + 11.6i)16-s + (49.1 − 85.0i)17-s + (53.0 + 91.7i)19-s + 49.4·20-s + 71.3·22-s + (27.1 + 47.0i)23-s + (−8.18 + 14.1i)25-s + (0.945 + 1.63i)26-s + ⋯
L(s)  = 1  + (0.346 + 0.600i)2-s + (0.259 − 0.450i)4-s + (0.531 + 0.920i)5-s + 1.05·8-s + (−0.368 + 0.638i)10-s + (0.499 − 0.864i)11-s + 0.0205·13-s + (0.104 + 0.181i)16-s + (0.700 − 1.21i)17-s + (0.639 + 1.10i)19-s + 0.552·20-s + 0.691·22-s + (0.246 + 0.426i)23-s + (−0.0654 + 0.113i)25-s + (0.00712 + 0.0123i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.827 - 0.561i$
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: \((2,\ 441,\ (\ :3/2),\ 0.827 - 0.561i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.187675084\)
\(L(\frac12)\) \(\approx\) \(3.187675084\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.979 - 1.69i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (-5.94 - 10.2i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-18.2 + 31.5i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 0.964T + 2.19e3T^{2} \)
17 \( 1 + (-49.1 + 85.0i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-53.0 - 91.7i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-27.1 - 47.0i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 229.T + 2.43e4T^{2} \)
31 \( 1 + (63.8 - 110. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (155. + 270. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 419.T + 6.89e4T^{2} \)
43 \( 1 - 523.T + 7.95e4T^{2} \)
47 \( 1 + (-135. - 234. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (125. - 217. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-204. + 353. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-430. - 745. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (253. - 438. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 523.T + 3.57e5T^{2} \)
73 \( 1 + (314. - 545. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (159. + 276. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 1.30e3T + 5.71e5T^{2} \)
89 \( 1 + (174. + 301. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 161.T + 9.12e5T^{2} \)
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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

−10.85106441084526151519611681754, −9.930593926976313200972581119786, −9.125625971307589118421791038686, −7.56481764539257577042792028611, −7.08905577605693207664929859755, −5.80563239185114670919968969713, −5.61533724366780679854981919122, −3.91671453483623232047081613470, −2.65550028925452378895434832939, −1.17907202897141682819239633469, 1.23006378399317224947516704563, 2.25955056058939396561618467280, 3.68454846549822764815947761169, 4.62118269643326141501814329267, 5.65614803403305765770828419641, 6.98073223920540015493509781377, 7.85944166797036572690472437488, 8.981794874200016877834983113361, 9.711103511929688460747526486141, 10.79324515965119251981220936446

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