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Properties

Label 2-21e2-147.131-c1-0-13
Degree $2$
Conductor $441$
Sign $0.500 + 0.865i$
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  + (1.72 − 0.677i)2-s + (1.05 − 0.975i)4-s + (0.820 − 0.559i)5-s + (0.202 − 2.63i)7-s + (−0.454 + 0.944i)8-s + (1.03 − 1.52i)10-s + (0.669 − 4.44i)11-s + (0.491 + 0.391i)13-s + (−1.43 − 4.68i)14-s + (−0.359 + 4.79i)16-s + (7.67 − 2.36i)17-s + (0.358 + 0.206i)19-s + (0.316 − 1.38i)20-s + (−1.85 − 8.11i)22-s + (−2.22 + 7.21i)23-s + ⋯
L(s)  = 1  + (1.21 − 0.478i)2-s + (0.525 − 0.487i)4-s + (0.366 − 0.250i)5-s + (0.0763 − 0.997i)7-s + (−0.160 + 0.333i)8-s + (0.327 − 0.480i)10-s + (0.201 − 1.33i)11-s + (0.136 + 0.108i)13-s + (−0.384 − 1.25i)14-s + (−0.0898 + 1.19i)16-s + (1.86 − 0.574i)17-s + (0.0821 + 0.0474i)19-s + (0.0708 − 0.310i)20-s + (−0.394 − 1.73i)22-s + (−0.463 + 1.50i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.500 + 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.500 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.500 + 0.865i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (278, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.500 + 0.865i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.30963 - 1.33187i\)
\(L(\frac12)\) \(\approx\) \(2.30963 - 1.33187i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-0.202 + 2.63i)T \)
good2 \( 1 + (-1.72 + 0.677i)T + (1.46 - 1.36i)T^{2} \)
5 \( 1 + (-0.820 + 0.559i)T + (1.82 - 4.65i)T^{2} \)
11 \( 1 + (-0.669 + 4.44i)T + (-10.5 - 3.24i)T^{2} \)
13 \( 1 + (-0.491 - 0.391i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (-7.67 + 2.36i)T + (14.0 - 9.57i)T^{2} \)
19 \( 1 + (-0.358 - 0.206i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.22 - 7.21i)T + (-19.0 - 12.9i)T^{2} \)
29 \( 1 + (0.523 + 0.119i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 + (7.47 - 4.31i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.17 + 4.79i)T + (2.76 + 36.8i)T^{2} \)
41 \( 1 + (-4.20 - 2.02i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (6.50 - 3.13i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (2.32 + 5.92i)T + (-34.4 + 31.9i)T^{2} \)
53 \( 1 + (-1.40 - 1.51i)T + (-3.96 + 52.8i)T^{2} \)
59 \( 1 + (-10.2 - 6.98i)T + (21.5 + 54.9i)T^{2} \)
61 \( 1 + (-3.28 + 3.53i)T + (-4.55 - 60.8i)T^{2} \)
67 \( 1 + (-1.75 - 3.03i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.14 - 0.488i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (-2.40 - 0.944i)T + (53.5 + 49.6i)T^{2} \)
79 \( 1 + (6.85 - 11.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.36 + 7.98i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-6.09 + 0.918i)T + (85.0 - 26.2i)T^{2} \)
97 \( 1 - 12.9iT - 97T^{2} \)
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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

−11.30506584453140149196529392195, −10.31837438366184597933231002361, −9.316115219668111802928548895455, −8.173029301971037800164752464992, −7.15201319177729671692948929041, −5.67976933854002537047196615579, −5.32517234960926873919851161459, −3.76007085412741258318320270323, −3.33531898348796276722640325436, −1.42954505120429003634281968697, 2.14316182946270853721662690113, 3.48903969828649412966933881229, 4.65404641640011899772074271415, 5.58693466106616025479064108385, 6.26492137807249676787896916982, 7.28161995676864245151535081311, 8.404723247191952324108023127636, 9.672263539323422406208523700394, 10.19711440342450046346342308118, 11.71542538514039548756098463731

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