Properties

Label 2-21e2-441.4-c1-0-29
Degree $2$
Conductor $441$
Sign $0.827 - 0.562i$
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  + (−0.162 + 2.16i)2-s + (0.451 − 1.67i)3-s + (−2.67 − 0.403i)4-s + (0.397 − 1.73i)5-s + (3.54 + 1.24i)6-s + (1.01 + 2.44i)7-s + (0.339 − 1.48i)8-s + (−2.59 − 1.51i)9-s + (3.69 + 1.14i)10-s + (3.38 − 1.63i)11-s + (−1.88 + 4.28i)12-s + (0.376 − 5.02i)13-s + (−5.44 + 1.80i)14-s + (−2.72 − 1.44i)15-s + (−2.00 − 0.617i)16-s + (6.99 − 1.05i)17-s + ⋯
L(s)  = 1  + (−0.114 + 1.52i)2-s + (0.260 − 0.965i)3-s + (−1.33 − 0.201i)4-s + (0.177 − 0.777i)5-s + (1.44 + 0.509i)6-s + (0.384 + 0.923i)7-s + (0.120 − 0.526i)8-s + (−0.863 − 0.503i)9-s + (1.16 + 0.360i)10-s + (1.02 − 0.492i)11-s + (−0.543 + 1.23i)12-s + (0.104 − 1.39i)13-s + (−1.45 + 0.482i)14-s + (−0.704 − 0.374i)15-s + (−0.500 − 0.154i)16-s + (1.69 − 0.255i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 - 0.562i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 - 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.827 - 0.562i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.827 - 0.562i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44525 + 0.444602i\)
\(L(\frac12)\) \(\approx\) \(1.44525 + 0.444602i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.451 + 1.67i)T \)
7 \( 1 + (-1.01 - 2.44i)T \)
good2 \( 1 + (0.162 - 2.16i)T + (-1.97 - 0.298i)T^{2} \)
5 \( 1 + (-0.397 + 1.73i)T + (-4.50 - 2.16i)T^{2} \)
11 \( 1 + (-3.38 + 1.63i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (-0.376 + 5.02i)T + (-12.8 - 1.93i)T^{2} \)
17 \( 1 + (-6.99 + 1.05i)T + (16.2 - 5.01i)T^{2} \)
19 \( 1 + (-0.0857 - 0.148i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.11 - 2.64i)T + (-5.11 - 22.4i)T^{2} \)
29 \( 1 + (-2.12 + 5.42i)T + (-21.2 - 19.7i)T^{2} \)
31 \( 1 + (-3.88 - 6.72i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.41 - 3.60i)T + (-27.1 - 25.1i)T^{2} \)
41 \( 1 + (-4.13 - 3.83i)T + (3.06 + 40.8i)T^{2} \)
43 \( 1 + (4.74 - 4.40i)T + (3.21 - 42.8i)T^{2} \)
47 \( 1 + (-0.823 + 10.9i)T + (-46.4 - 7.00i)T^{2} \)
53 \( 1 + (3.34 + 8.52i)T + (-38.8 + 36.0i)T^{2} \)
59 \( 1 + (0.247 - 0.229i)T + (4.40 - 58.8i)T^{2} \)
61 \( 1 + (8.24 - 1.24i)T + (58.2 - 17.9i)T^{2} \)
67 \( 1 + (3.41 + 5.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.15 - 3.96i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (12.0 - 8.23i)T + (26.6 - 67.9i)T^{2} \)
79 \( 1 + (-1.63 + 2.82i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.369 - 4.92i)T + (-82.0 + 12.3i)T^{2} \)
89 \( 1 + (-0.394 - 5.26i)T + (-88.0 + 13.2i)T^{2} \)
97 \( 1 + (0.624 + 1.08i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.63136959801527767985722228842, −9.827714350562520757696022007393, −8.822383224264354132644830519893, −8.267788026895437942060188844002, −7.70614889272851538157307007816, −6.47111701862843253638092248389, −5.72760323863506458365316148628, −5.10362339652579732253378667492, −3.11251210349018543690770009551, −1.18801968105586071219762345774, 1.56494088890596455833931653099, 2.94647672430865254239335358562, 3.95891355478986602563053110029, 4.51740492915774517818971996060, 6.30436284175922908247920175849, 7.49063151830363984088186473991, 8.874609687876184545450862174790, 9.610452484608592052579913479708, 10.33219282772698104369517536088, 10.82904821499369109680444557570

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