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Properties

Label 2-21e2-21.17-c3-0-9
Degree $2$
Conductor $441$
Sign $-0.708 - 0.706i$
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  + (0.555 − 0.320i)2-s + (−3.79 + 6.57i)4-s + (6.03 + 10.4i)5-s + 9.99i·8-s + (6.70 + 3.86i)10-s + (38.0 + 21.9i)11-s + 66.9i·13-s + (−27.1 − 47.0i)16-s + (19.7 − 34.1i)17-s + (50.1 − 28.9i)19-s − 91.6·20-s + 28.1·22-s + (−39.9 + 23.0i)23-s + (−10.3 + 17.9i)25-s + (21.4 + 37.1i)26-s + ⋯
L(s)  = 1  + (0.196 − 0.113i)2-s + (−0.474 + 0.821i)4-s + (0.539 + 0.935i)5-s + 0.441i·8-s + (0.211 + 0.122i)10-s + (1.04 + 0.602i)11-s + 1.42i·13-s + (−0.424 − 0.734i)16-s + (0.281 − 0.487i)17-s + (0.605 − 0.349i)19-s − 1.02·20-s + 0.272·22-s + (−0.362 + 0.209i)23-s + (−0.0830 + 0.143i)25-s + (0.161 + 0.280i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.708 - 0.706i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.708 - 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.708 - 0.706i$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ -0.708 - 0.706i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.868378197\)
\(L(\frac12)\) \(\approx\) \(1.868378197\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.555 + 0.320i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + (-6.03 - 10.4i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-38.0 - 21.9i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 66.9iT - 2.19e3T^{2} \)
17 \( 1 + (-19.7 + 34.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-50.1 + 28.9i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (39.9 - 23.0i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 201. iT - 2.43e4T^{2} \)
31 \( 1 + (160. + 92.7i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-205. - 355. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 408.T + 6.89e4T^{2} \)
43 \( 1 - 129.T + 7.95e4T^{2} \)
47 \( 1 + (257. + 445. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (378. + 218. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-303. + 526. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (21.2 - 12.2i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (220. - 382. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 1.15e3iT - 3.57e5T^{2} \)
73 \( 1 + (-454. - 262. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-75.9 - 131. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 1.30e3T + 5.71e5T^{2} \)
89 \( 1 + (-291. - 505. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.09e3iT - 9.12e5T^{2} \)
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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

−11.41740769420605067083682647196, −9.908277051364545364595920613076, −9.392718954654174843200041129631, −8.394041944101565179309247578817, −7.04575075573533122859472356116, −6.71402971510748531236068760427, −5.15580649664147129989352369188, −4.07582269841030158352317640774, −3.08381341845956622823880685562, −1.80958085171548987054946321400, 0.59741168432814101932053452995, 1.56350662526012174266331389043, 3.51459983328016406339855385216, 4.68844136647446222743226633676, 5.71426944866332548775463013615, 6.10242035232252929008898253396, 7.72113002357850618973435677637, 8.757811920646721117152713057190, 9.440078286950399953978806609168, 10.21122639779253745418329449351

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