Properties

Label 2-21e2-441.142-c1-0-3
Degree $2$
Conductor $441$
Sign $-0.611 + 0.791i$
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  + (−0.643 + 1.63i)2-s + (−1.73 − 0.0118i)3-s + (−0.805 − 0.747i)4-s + (0.303 − 0.146i)5-s + (1.13 − 2.83i)6-s + (0.198 + 2.63i)7-s + (−1.42 + 0.688i)8-s + (2.99 + 0.0411i)9-s + (0.0442 + 0.591i)10-s + (−0.238 − 0.299i)11-s + (1.38 + 1.30i)12-s + (−1.09 + 2.78i)13-s + (−4.45 − 1.37i)14-s + (−0.527 + 0.249i)15-s + (−0.372 − 4.97i)16-s + (−4.16 + 3.86i)17-s + ⋯
L(s)  = 1  + (−0.454 + 1.15i)2-s + (−0.999 − 0.00686i)3-s + (−0.402 − 0.373i)4-s + (0.135 − 0.0653i)5-s + (0.462 − 1.15i)6-s + (0.0750 + 0.997i)7-s + (−0.505 + 0.243i)8-s + (0.999 + 0.0137i)9-s + (0.0140 + 0.186i)10-s + (−0.0719 − 0.0902i)11-s + (0.400 + 0.376i)12-s + (−0.303 + 0.773i)13-s + (−1.18 − 0.366i)14-s + (−0.136 + 0.0644i)15-s + (−0.0932 − 1.24i)16-s + (−1.00 + 0.936i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.611 + 0.791i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.611 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.611 + 0.791i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.611 + 0.791i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.169169 - 0.344466i\)
\(L(\frac12)\) \(\approx\) \(0.169169 - 0.344466i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.73 + 0.0118i)T \)
7 \( 1 + (-0.198 - 2.63i)T \)
good2 \( 1 + (0.643 - 1.63i)T + (-1.46 - 1.36i)T^{2} \)
5 \( 1 + (-0.303 + 0.146i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (0.238 + 0.299i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (1.09 - 2.78i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (4.16 - 3.86i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (0.105 - 0.182i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.89 + 8.28i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (8.58 + 2.64i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (-1.34 + 2.32i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.63 - 2.04i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (7.86 + 5.36i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (-6.35 + 4.33i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (4.13 - 10.5i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (0.444 - 0.137i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (-4.01 + 2.73i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (3.45 - 3.20i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (0.224 - 0.388i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.87 - 8.23i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (14.1 + 2.13i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (-3.17 - 5.50i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.16 - 13.1i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (-2.02 - 5.14i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (5.67 - 9.82i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.62841792011061088599669091959, −10.91185823833263733665160158525, −9.600809205148310194485328988153, −8.918051141493113694562011198228, −7.950412837020629807223063590325, −6.87245443985924262805181574694, −6.17993422269186228158863765427, −5.48271323529177296291813980217, −4.33383086775340677871764842225, −2.21222814256630595819584477328, 0.30611008479507886421231402476, 1.70674524230519414582864466416, 3.31193055324295712583144693091, 4.53573529237487555541538840529, 5.70493909615548953341975383803, 6.85168176478911884488212010066, 7.69046652160514498422452507574, 9.270819153684546370935617860499, 9.941715940865113277102742516546, 10.64395316943296119096247944126

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