↑

Properties

Label 2-21e2-63.38-c1-0-25
Degree $2$
Conductor $441$
Sign $0.253 + 0.967i$
Analytic cond. $3.52140$
Root an. cond. $1.87654$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.122i·2-s + (−0.937 − 1.45i)3-s + 1.98·4-s + (0.264 − 0.458i)5-s + (−0.178 + 0.114i)6-s − 0.487i·8-s + (−1.24 + 2.73i)9-s + (−0.0560 − 0.0323i)10-s + (3.64 − 2.10i)11-s + (−1.86 − 2.89i)12-s + (1.74 − 1.00i)13-s + (−0.915 + 0.0444i)15-s + 3.91·16-s + (2.19 − 3.79i)17-s + (0.334 + 0.151i)18-s + (−4.54 + 2.62i)19-s + ⋯
L(s)  = 1  − 0.0865i·2-s + (−0.541 − 0.840i)3-s + 0.992·4-s + (0.118 − 0.205i)5-s + (−0.0727 + 0.0468i)6-s − 0.172i·8-s + (−0.413 + 0.910i)9-s + (−0.0177 − 0.0102i)10-s + (1.09 − 0.633i)11-s + (−0.537 − 0.834i)12-s + (0.484 − 0.279i)13-s + (−0.236 + 0.0114i)15-s + 0.977·16-s + (0.532 − 0.921i)17-s + (0.0787 + 0.0358i)18-s + (−1.04 + 0.601i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.253 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.253 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.253 + 0.967i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.253 + 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21163 - 0.935008i\)
\(L(\frac12)\) \(\approx\) \(1.21163 - 0.935008i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.937 + 1.45i)T \)
7 \( 1 \)
good2 \( 1 + 0.122iT - 2T^{2} \)
5 \( 1 + (-0.264 + 0.458i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.64 + 2.10i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.74 + 1.00i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.19 + 3.79i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.54 - 2.62i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.43 + 3.13i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (7.27 + 4.20i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.19iT - 31T^{2} \)
37 \( 1 + (-1.61 - 2.79i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.0994 - 0.172i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.96 + 6.86i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 9.97T + 47T^{2} \)
53 \( 1 + (-3.65 - 2.10i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 13.4T + 59T^{2} \)
61 \( 1 - 13.1iT - 61T^{2} \)
67 \( 1 + 6.58T + 67T^{2} \)
71 \( 1 - 8.50iT - 71T^{2} \)
73 \( 1 + (-4.86 - 2.80i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 0.572T + 79T^{2} \)
83 \( 1 + (5.42 - 9.39i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.43 - 11.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.493 + 0.285i)T + (48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.09931589358254409472428691937, −10.36360521689955672152180168026, −9.035085825721115411291705192471, −7.973071457723733667391932932611, −7.14625538321469060838335687339, −6.15453164752203169235912075270, −5.68157286307946314460810994279, −3.89504035088653039544115481801, −2.39944454021182578651782880135, −1.14123166509193816426109444159, 1.80471087565728098913132466129, 3.47053563898301788791896491065, 4.41693884929319407707494381735, 5.93266657190433183776340298324, 6.36492305953321046519618931579, 7.44909088640768016929848336223, 8.759208617061139758212584897383, 9.663897647753540270147647378967, 10.63044783463003724664430190755, 11.10114486994123231242759116854

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