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Properties

Label 2-21e2-49.8-c1-0-10
Degree $2$
Conductor $441$
Sign $0.808 - 0.588i$
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  + (0.894 + 1.12i)2-s + (−0.0127 + 0.0560i)4-s + (−1.03 + 0.499i)5-s + (2.02 − 1.70i)7-s + (2.51 − 1.20i)8-s + (−1.48 − 0.716i)10-s + (3.40 + 4.27i)11-s + (−2.42 − 3.04i)13-s + (3.72 + 0.748i)14-s + (3.70 + 1.78i)16-s + (−0.517 − 2.26i)17-s + 5.93·19-s + (−0.0147 − 0.0644i)20-s + (−1.74 + 7.64i)22-s + (−1.30 + 5.71i)23-s + ⋯
L(s)  = 1  + (0.632 + 0.792i)2-s + (−0.00639 + 0.0280i)4-s + (−0.463 + 0.223i)5-s + (0.765 − 0.643i)7-s + (0.887 − 0.427i)8-s + (−0.470 − 0.226i)10-s + (1.02 + 1.28i)11-s + (−0.673 − 0.844i)13-s + (0.994 + 0.199i)14-s + (0.926 + 0.445i)16-s + (−0.125 − 0.550i)17-s + 1.36·19-s + (−0.00329 − 0.0144i)20-s + (−0.372 + 1.63i)22-s + (−0.271 + 1.19i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.808 - 0.588i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.808 - 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.808 - 0.588i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.808 - 0.588i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.00416 + 0.651854i\)
\(L(\frac12)\) \(\approx\) \(2.00416 + 0.651854i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.02 + 1.70i)T \)
good2 \( 1 + (-0.894 - 1.12i)T + (-0.445 + 1.94i)T^{2} \)
5 \( 1 + (1.03 - 0.499i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (-3.40 - 4.27i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (2.42 + 3.04i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + (0.517 + 2.26i)T + (-15.3 + 7.37i)T^{2} \)
19 \( 1 - 5.93T + 19T^{2} \)
23 \( 1 + (1.30 - 5.71i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (1.30 + 5.70i)T + (-26.1 + 12.5i)T^{2} \)
31 \( 1 - 3.41T + 31T^{2} \)
37 \( 1 + (-1.54 - 6.74i)T + (-33.3 + 16.0i)T^{2} \)
41 \( 1 + (8.31 - 4.00i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (3.93 + 1.89i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (4.53 + 5.68i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (-1.38 + 6.04i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 + (3.94 + 1.89i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (0.988 + 4.33i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 + 12.5T + 67T^{2} \)
71 \( 1 + (0.126 - 0.554i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (9.13 - 11.4i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 - 9.54T + 79T^{2} \)
83 \( 1 + (2.58 - 3.23i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (-1.54 + 1.93i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 - 11.7T + 97T^{2} \)
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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

−11.55317466649529410478397568484, −10.07946505005164636793842252541, −9.708650865257284593782970607309, −7.922139264331556559669622866095, −7.41859331168766332748825823046, −6.69938334205157812757049934271, −5.31781166506515681503942022327, −4.65167759677249169908128117391, −3.57981410719017328583080584006, −1.51153018690973650942715865090, 1.59810822141970454448060026998, 2.97785909060540725315226670173, 4.07457047132291604849082169286, 4.91303077374430655127831640887, 6.13031628536300414337675175120, 7.46798137406473340024132776646, 8.412192334183041712612379050003, 9.111865207515428267492913173446, 10.51587547237966666771121886247, 11.38775567802413112948852284090

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