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Properties

Label 2-21e2-7.4-c3-0-14
Degree $2$
Conductor $441$
Sign $-0.266 - 0.963i$
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  + (1.13 + 1.97i)2-s + (1.41 − 2.44i)4-s + (2.27 + 3.94i)5-s + 24.6·8-s + (−5.17 + 8.96i)10-s + (−20.3 + 35.2i)11-s − 53.2·13-s + (16.7 + 28.9i)16-s + (−2.27 + 3.94i)17-s + (61.2 + 106. i)19-s + 12.8·20-s − 92.7·22-s + (65.6 + 113. i)23-s + (52.1 − 90.3i)25-s + (−60.6 − 105. i)26-s + ⋯
L(s)  = 1  + (0.402 + 0.696i)2-s + (0.176 − 0.305i)4-s + (0.203 + 0.352i)5-s + 1.08·8-s + (−0.163 + 0.283i)10-s + (−0.558 + 0.967i)11-s − 1.13·13-s + (0.261 + 0.452i)16-s + (−0.0324 + 0.0562i)17-s + (0.740 + 1.28i)19-s + 0.143·20-s − 0.898·22-s + (0.595 + 1.03i)23-s + (0.417 − 0.722i)25-s + (−0.457 − 0.792i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.266 - 0.963i$
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: \((2,\ 441,\ (\ :3/2),\ -0.266 - 0.963i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.462118322\)
\(L(\frac12)\) \(\approx\) \(2.462118322\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.13 - 1.97i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (-2.27 - 3.94i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (20.3 - 35.2i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 53.2T + 2.19e3T^{2} \)
17 \( 1 + (2.27 - 3.94i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-61.2 - 106. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-65.6 - 113. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 216.T + 2.43e4T^{2} \)
31 \( 1 + (125. - 218. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (5.94 + 10.3i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 111.T + 6.89e4T^{2} \)
43 \( 1 - 369.T + 7.95e4T^{2} \)
47 \( 1 + (-131. - 227. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (283. - 491. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (419. - 727. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (242. + 420. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-166. + 288. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 590.T + 3.57e5T^{2} \)
73 \( 1 + (-245. + 424. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (60.8 + 105. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 609.T + 5.71e5T^{2} \)
89 \( 1 + (359. + 622. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 637.T + 9.12e5T^{2} \)
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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

−10.66350431011661844612258605515, −10.22293616145755157301334456712, −9.291315969579068289128215888818, −7.72199465518980645181231423431, −7.29737309779768208080400463177, −6.26672660076163992467146983176, −5.27997450529195642963789830790, −4.52653416079034541233277465235, −2.84666450946978389468064001712, −1.52423803825939629928149785508, 0.70217724294105440727984128171, 2.38946700980819698518632351688, 3.14946695947085764736892777683, 4.56835847267012128274196257061, 5.31383121792435877150821514093, 6.79434866212152922347428695626, 7.67040300192020027748427169652, 8.674257210749041339583398489946, 9.667137600233792979736240245241, 10.73437376270562243017201261502

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