Properties

Label 2-21e2-441.142-c1-0-47
Degree $2$
Conductor $441$
Sign $-0.999 - 0.0412i$
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  + (0.768 − 1.95i)2-s + (−1.69 − 0.353i)3-s + (−1.77 − 1.64i)4-s + (1.62 − 0.782i)5-s + (−1.99 + 3.04i)6-s + (0.533 − 2.59i)7-s + (−0.796 + 0.383i)8-s + (2.74 + 1.19i)9-s + (−0.283 − 3.78i)10-s + (−2.81 − 3.52i)11-s + (2.42 + 3.41i)12-s + (−1.23 + 3.14i)13-s + (−4.66 − 3.03i)14-s + (−3.03 + 0.751i)15-s + (−0.222 − 2.97i)16-s + (1.00 − 0.934i)17-s + ⋯
L(s)  = 1  + (0.543 − 1.38i)2-s + (−0.978 − 0.204i)3-s + (−0.887 − 0.823i)4-s + (0.726 − 0.349i)5-s + (−0.814 + 1.24i)6-s + (0.201 − 0.979i)7-s + (−0.281 + 0.135i)8-s + (0.916 + 0.399i)9-s + (−0.0895 − 1.19i)10-s + (−0.847 − 1.06i)11-s + (0.700 + 0.986i)12-s + (−0.341 + 0.871i)13-s + (−1.24 − 0.811i)14-s + (−0.782 + 0.194i)15-s + (−0.0557 − 0.743i)16-s + (0.244 − 0.226i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.999 - 0.0412i$
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: \((2,\ 441,\ (\ :1/2),\ -0.999 - 0.0412i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0290428 + 1.40674i\)
\(L(\frac12)\) \(\approx\) \(0.0290428 + 1.40674i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.69 + 0.353i)T \)
7 \( 1 + (-0.533 + 2.59i)T \)
good2 \( 1 + (-0.768 + 1.95i)T + (-1.46 - 1.36i)T^{2} \)
5 \( 1 + (-1.62 + 0.782i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (2.81 + 3.52i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (1.23 - 3.14i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (-1.00 + 0.934i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (0.818 - 1.41i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.0235 + 0.103i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (-4.37 - 1.35i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (0.961 - 1.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (7.36 + 2.27i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (-6.11 - 4.16i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (-7.79 + 5.31i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (3.71 - 9.47i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (-6.04 + 1.86i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (-9.90 + 6.75i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (-9.55 + 8.86i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (-2.44 + 4.24i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.11 + 13.6i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (3.92 + 0.591i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (0.102 + 0.176i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.41 - 11.2i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (-2.66 - 6.79i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (2.46 - 4.27i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.84291630011456386123328437273, −10.23316705520752779902905787834, −9.382995069314684391768015544607, −7.84299008692411961860292801676, −6.73438953352656766693732479514, −5.50084280852771582400685596721, −4.77260097948113935584177406306, −3.66604547818177725502061240727, −2.08696652164492376833526427502, −0.867691994122177211982222467133, 2.33220239905289007303671964779, 4.36963336451930566911793430404, 5.41177471786773028250630832135, 5.67014338937497642351519253786, 6.73562273327658266804584247012, 7.52586859640743559962596248972, 8.616204404931425625453876476326, 9.974713341721406807478944386720, 10.45193054017549187326252503890, 11.74110303047258584092089723297

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