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Properties

Label 2-21e2-63.38-c1-0-8
Degree $2$
Conductor $441$
Sign $-0.953 + 0.300i$
Analytic cond. $3.52140$
Root an. cond. $1.87654$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.37i·2-s + (1.65 − 0.523i)3-s − 3.62·4-s + (−1.71 + 2.97i)5-s + (1.24 + 3.91i)6-s − 3.86i·8-s + (2.45 − 1.72i)9-s + (−7.05 − 4.07i)10-s + (−0.271 + 0.156i)11-s + (−5.99 + 1.90i)12-s + (−5.09 + 2.94i)13-s + (−1.27 + 5.81i)15-s + 1.91·16-s + (−0.476 + 0.825i)17-s + (4.10 + 5.81i)18-s + (1.09 − 0.630i)19-s + ⋯
L(s)  = 1  + 1.67i·2-s + (0.953 − 0.302i)3-s − 1.81·4-s + (−0.768 + 1.33i)5-s + (0.507 + 1.59i)6-s − 1.36i·8-s + (0.817 − 0.576i)9-s + (−2.23 − 1.28i)10-s + (−0.0819 + 0.0473i)11-s + (−1.73 + 0.548i)12-s + (−1.41 + 0.816i)13-s + (−0.329 + 1.50i)15-s + 0.479·16-s + (−0.115 + 0.200i)17-s + (0.967 + 1.37i)18-s + (0.250 − 0.144i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 + 0.300i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.953 + 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.953 + 0.300i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.953 + 0.300i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.199804 - 1.29906i\)
\(L(\frac12)\) \(\approx\) \(0.199804 - 1.29906i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.65 + 0.523i)T \)
7 \( 1 \)
good2 \( 1 - 2.37iT - 2T^{2} \)
5 \( 1 + (1.71 - 2.97i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.271 - 0.156i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (5.09 - 2.94i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.476 - 0.825i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.09 + 0.630i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.91 - 3.41i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.43 - 1.98i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.23iT - 31T^{2} \)
37 \( 1 + (2.68 + 4.65i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.0699 - 0.121i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.44 + 2.49i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 2.01T + 47T^{2} \)
53 \( 1 + (-10.3 - 5.98i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 1.64T + 59T^{2} \)
61 \( 1 + 2.97iT - 61T^{2} \)
67 \( 1 + 1.86T + 67T^{2} \)
71 \( 1 - 10.9iT - 71T^{2} \)
73 \( 1 + (-0.354 - 0.204i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 10.4T + 79T^{2} \)
83 \( 1 + (4.00 - 6.92i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.05 - 1.83i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.5 - 6.06i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.71203397509007423710933245010, −10.47474508402520456420232056661, −9.374827875975721893196656842773, −8.619242916736947534960123661950, −7.43969257377002939875072478090, −7.25132151091283755444147670243, −6.56469664007355284852478169956, −5.03483533775395751511129814788, −3.87307568670106156863760805466, −2.64144087984720413622083121468, 0.76237576638491662765942189928, 2.36708662750943439512364387978, 3.36582333455930235393356695360, 4.52510494162901125218180745378, 4.97658069601018665257870067523, 7.41641527926450518609976112887, 8.335775989004786485826876329619, 9.028662689126578501183040206582, 9.802948109075374757744227407811, 10.52784949335044624729260501432

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