Properties

Label 2-21e2-21.5-c3-0-2
Degree $2$
Conductor $441$
Sign $-0.851 - 0.524i$
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  + (0.648 + 0.374i)2-s + (−3.71 − 6.44i)4-s + (−5.42 + 9.40i)5-s − 11.5i·8-s + (−7.04 + 4.06i)10-s + (44.9 − 25.9i)11-s + 32.1i·13-s + (−25.4 + 44.0i)16-s + (−40.7 − 70.5i)17-s + (0.0420 + 0.0242i)19-s + 80.7·20-s + 38.8·22-s + (−77.3 − 44.6i)23-s + (3.58 + 6.20i)25-s + (−12.0 + 20.8i)26-s + ⋯
L(s)  = 1  + (0.229 + 0.132i)2-s + (−0.464 − 0.805i)4-s + (−0.485 + 0.840i)5-s − 0.511i·8-s + (−0.222 + 0.128i)10-s + (1.23 − 0.710i)11-s + 0.686i·13-s + (−0.397 + 0.688i)16-s + (−0.581 − 1.00i)17-s + (0.000507 + 0.000293i)19-s + 0.902·20-s + 0.376·22-s + (−0.701 − 0.404i)23-s + (0.0286 + 0.0496i)25-s + (−0.0909 + 0.157i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.851 - 0.524i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.851 - 0.524i$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ -0.851 - 0.524i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3987305894\)
\(L(\frac12)\) \(\approx\) \(0.3987305894\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.648 - 0.374i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + (5.42 - 9.40i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-44.9 + 25.9i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 32.1iT - 2.19e3T^{2} \)
17 \( 1 + (40.7 + 70.5i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-0.0420 - 0.0242i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (77.3 + 44.6i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 175. iT - 2.43e4T^{2} \)
31 \( 1 + (186. - 107. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (32.2 - 55.8i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 411.T + 6.89e4T^{2} \)
43 \( 1 + 234.T + 7.95e4T^{2} \)
47 \( 1 + (316. - 547. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-230. + 132. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (175. + 304. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-673. - 389. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-98.0 - 169. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 142. iT - 3.57e5T^{2} \)
73 \( 1 + (676. - 390. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (644. - 1.11e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 235.T + 5.71e5T^{2} \)
89 \( 1 + (-335. + 580. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 655. iT - 9.12e5T^{2} \)
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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

−11.19966270827511483208582249610, −10.19885739332900455361819427475, −9.252815939354012056827034936196, −8.551125670624703446794158422335, −6.93146457020520851550880864635, −6.62719546442712906632239063265, −5.35348160639783846297580178347, −4.23923676389200771664246162606, −3.26582619948167570863236682143, −1.49094912627312242960532741941, 0.12287582739852679291358856690, 1.88931978382610265040616871670, 3.66123575150013052693593512778, 4.21962898197303518197294992215, 5.27598999819915751569339507771, 6.63802522956013949854373448536, 7.81235050651168061823543096093, 8.480024992624941482749993102183, 9.241844562966691045816757085139, 10.27387416794512321875121349109

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