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Properties

Label 2-21e2-21.17-c3-0-12
Degree $2$
Conductor $441$
Sign $-0.0285 - 0.999i$
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  + (−4.68 + 2.70i)2-s + (10.6 − 18.3i)4-s + 71.5i·8-s + (57.8 + 33.3i)11-s + (−108. − 187. i)16-s − 361.·22-s + (−108. + 62.5i)23-s + (62.5 − 108. i)25-s − 69.7i·29-s + (519. + 300. i)32-s + (−5.29 − 9.16i)37-s + 534.·43-s + (1.22e3 − 708. i)44-s + (338. − 585. i)46-s + 675. i·50-s + ⋯
L(s)  = 1  + (−1.65 + 0.955i)2-s + (1.32 − 2.29i)4-s + 3.16i·8-s + (1.58 + 0.915i)11-s + (−1.69 − 2.93i)16-s − 3.49·22-s + (−0.982 + 0.567i)23-s + (0.5 − 0.866i)25-s − 0.446i·29-s + (2.87 + 1.65i)32-s + (−0.0235 − 0.0407i)37-s + 1.89·43-s + (4.20 − 2.42i)44-s + (1.08 − 1.87i)46-s + 1.91i·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0285 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0285 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.0285 - 0.999i$
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: \((2,\ 441,\ (\ :3/2),\ -0.0285 - 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8558672709\)
\(L(\frac12)\) \(\approx\) \(0.8558672709\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.68 - 2.70i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-57.8 - 33.3i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 2.19e3T^{2} \)
17 \( 1 + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (108. - 62.5i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 69.7iT - 2.43e4T^{2} \)
31 \( 1 + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (5.29 + 9.16i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 6.89e4T^{2} \)
43 \( 1 - 534.T + 7.95e4T^{2} \)
47 \( 1 + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-56.7 - 32.7i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (370 - 640. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 205. iT - 3.57e5T^{2} \)
73 \( 1 + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-692 - 1.19e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 5.71e5T^{2} \)
89 \( 1 + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 9.12e5T^{2} \)
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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

−10.53443364593819821199042356931, −9.687101102833158799947235125850, −9.163680953601587834021325317758, −8.231475226221811495662112767153, −7.32494090745422597224030915248, −6.58496473115298367566927297947, −5.76562972391473422723312738435, −4.28538037042091896824221154394, −2.11605828934907865132127737644, −0.963280809221571099333487696531, 0.64530178916467290401507709622, 1.71159953722372942780615816214, 3.09109051484486819711508720021, 4.04829320467005465304248555635, 6.14226242912219829074560788871, 7.08792587872193322274574322429, 8.096199056377145089319766885176, 8.964315392398677144081690304669, 9.396903078946578868705494563414, 10.51297544396566276101716412236

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