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Properties

Label 2-21e2-7.2-c3-0-42
Degree $2$
Conductor $441$
Sign $-0.605 + 0.795i$
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  + (0.799 − 1.38i)2-s + (2.72 + 4.71i)4-s + (9.14 − 15.8i)5-s + 21.4·8-s + (−14.6 − 25.3i)10-s + (−30.6 − 53.0i)11-s − 32.4·13-s + (−4.61 + 7.99i)16-s + (−40.6 − 70.4i)17-s + (−10.4 + 18.1i)19-s + 99.6·20-s − 97.9·22-s + (16.8 − 29.2i)23-s + (−104. − 181. i)25-s + (−25.9 + 44.9i)26-s + ⋯
L(s)  = 1  + (0.282 − 0.489i)2-s + (0.340 + 0.589i)4-s + (0.818 − 1.41i)5-s + 0.949·8-s + (−0.462 − 0.800i)10-s + (−0.839 − 1.45i)11-s − 0.692·13-s + (−0.0721 + 0.124i)16-s + (−0.580 − 1.00i)17-s + (−0.126 + 0.218i)19-s + 1.11·20-s − 0.948·22-s + (0.152 − 0.264i)23-s + (−0.838 − 1.45i)25-s + (−0.195 + 0.338i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.605 + 0.795i$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ -0.605 + 0.795i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.362057679\)
\(L(\frac12)\) \(\approx\) \(2.362057679\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.799 + 1.38i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + (-9.14 + 15.8i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (30.6 + 53.0i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 32.4T + 2.19e3T^{2} \)
17 \( 1 + (40.6 + 70.4i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (10.4 - 18.1i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-16.8 + 29.2i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 52.0T + 2.43e4T^{2} \)
31 \( 1 + (-96.9 - 167. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-133. + 231. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 203.T + 6.89e4T^{2} \)
43 \( 1 + 21.9T + 7.95e4T^{2} \)
47 \( 1 + (123. - 214. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (70.4 + 121. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (110. + 191. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-326. + 565. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (302. + 523. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 716.T + 3.57e5T^{2} \)
73 \( 1 + (-194. - 336. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-144. + 250. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 115.T + 5.71e5T^{2} \)
89 \( 1 + (-469. + 813. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 120.T + 9.12e5T^{2} \)
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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

−10.52291349025807522111677817186, −9.455921124236133780543868126337, −8.567209259809423708526199716074, −7.87365454708665884996167052097, −6.53144630346208697358740652643, −5.29890325580181804573562915679, −4.64793408210023657437691635337, −3.15265786417673857755342412179, −2.10202303477560897939539403830, −0.64054517918617182664615303989, 1.90740260159522533090452093968, 2.63801065460833442558346961846, 4.46130123303381068859719101752, 5.48562730512313905884064334173, 6.49202446545260353160715361130, 7.00378934375320765696362075133, 7.910576651966954999505551378421, 9.649705489152691287276023184350, 10.18384063599934923169708576336, 10.71694730024413937347715222727

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