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Properties

Label 2-21e2-441.101-c1-0-24
Degree $2$
Conductor $441$
Sign $0.936 + 0.351i$
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  + (0.787 − 0.627i)2-s + (−1.00 − 1.41i)3-s + (−0.219 + 0.961i)4-s + (−0.00591 + 0.0788i)5-s + (−1.67 − 0.480i)6-s + (2.62 + 0.367i)7-s + (1.30 + 2.70i)8-s + (−0.981 + 2.83i)9-s + (0.0448 + 0.0657i)10-s + (4.74 + 1.86i)11-s + (1.57 − 0.656i)12-s + (0.190 + 0.0746i)13-s + (2.29 − 1.35i)14-s + (0.117 − 0.0708i)15-s + (0.949 + 0.457i)16-s + (−6.52 − 2.01i)17-s + ⋯
L(s)  = 1  + (0.556 − 0.443i)2-s + (−0.579 − 0.814i)3-s + (−0.109 + 0.480i)4-s + (−0.00264 + 0.0352i)5-s + (−0.684 − 0.195i)6-s + (0.990 + 0.138i)7-s + (0.461 + 0.957i)8-s + (−0.327 + 0.944i)9-s + (0.0141 + 0.0208i)10-s + (1.42 + 0.561i)11-s + (0.455 − 0.189i)12-s + (0.0527 + 0.0207i)13-s + (0.612 − 0.362i)14-s + (0.0302 − 0.0183i)15-s + (0.237 + 0.114i)16-s + (−1.58 − 0.488i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.351i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 + 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.936 + 0.351i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.936 + 0.351i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69822 - 0.308022i\)
\(L(\frac12)\) \(\approx\) \(1.69822 - 0.308022i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.00 + 1.41i)T \)
7 \( 1 + (-2.62 - 0.367i)T \)
good2 \( 1 + (-0.787 + 0.627i)T + (0.445 - 1.94i)T^{2} \)
5 \( 1 + (0.00591 - 0.0788i)T + (-4.94 - 0.745i)T^{2} \)
11 \( 1 + (-4.74 - 1.86i)T + (8.06 + 7.48i)T^{2} \)
13 \( 1 + (-0.190 - 0.0746i)T + (9.52 + 8.84i)T^{2} \)
17 \( 1 + (6.52 + 2.01i)T + (14.0 + 9.57i)T^{2} \)
19 \( 1 + (-1.26 + 0.731i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.65 + 6.09i)T + (-1.71 - 22.9i)T^{2} \)
29 \( 1 + (-1.19 + 3.86i)T + (-23.9 - 16.3i)T^{2} \)
31 \( 1 - 7.08iT - 31T^{2} \)
37 \( 1 + (1.30 - 1.20i)T + (2.76 - 36.8i)T^{2} \)
41 \( 1 + (-1.88 - 1.28i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (7.26 - 4.95i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (1.21 + 1.52i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (2.35 - 2.54i)T + (-3.96 - 52.8i)T^{2} \)
59 \( 1 + (9.77 + 4.70i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (-13.9 + 3.19i)T + (54.9 - 26.4i)T^{2} \)
67 \( 1 + 10.2T + 67T^{2} \)
71 \( 1 + (-0.0500 - 0.0114i)T + (63.9 + 30.8i)T^{2} \)
73 \( 1 + (7.84 - 3.07i)T + (53.5 - 49.6i)T^{2} \)
79 \( 1 + 1.86T + 79T^{2} \)
83 \( 1 + (3.83 + 9.77i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (-10.9 - 1.64i)T + (85.0 + 26.2i)T^{2} \)
97 \( 1 + (-1.21 - 0.702i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.36992073948497753401155279771, −10.73187941839691051918884115778, −8.987077422714164720187265254427, −8.384122289927979477372557719853, −7.15438743691657016623851449784, −6.55570804116245226995186300917, −4.91345500533279141069956631718, −4.51775619093573824322472290424, −2.76399360282676994403192398257, −1.55161933254377523573419209104, 1.25293187933289473860912834068, 3.70796987226027066831865100313, 4.52663626778528183031841527873, 5.33326083972861806290317633074, 6.29068930792993246135181308245, 7.08172313863601181329152733035, 8.757718911205655774555067980527, 9.275053329555948495454439262622, 10.48631868439972108985932181549, 11.16102477278546639333868351079

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