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Properties

Label 2-21e2-63.41-c1-0-32
Degree $2$
Conductor $441$
Sign $-0.445 + 0.895i$
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  + (2.23 − 1.28i)2-s + (−1.71 + 0.266i)3-s + (2.32 − 4.02i)4-s + (1.16 − 2.01i)5-s + (−3.47 + 2.80i)6-s − 6.82i·8-s + (2.85 − 0.911i)9-s − 6.01i·10-s + (−3.78 + 2.18i)11-s + (−2.90 + 7.50i)12-s + (−1.14 − 0.660i)13-s + (−1.45 + 3.76i)15-s + (−4.15 − 7.18i)16-s + 5.78·17-s + (5.20 − 5.71i)18-s − 0.675i·19-s + ⋯
L(s)  = 1  + (1.57 − 0.911i)2-s + (−0.988 + 0.153i)3-s + (1.16 − 2.01i)4-s + (0.521 − 0.903i)5-s + (−1.41 + 1.14i)6-s − 2.41i·8-s + (0.952 − 0.303i)9-s − 1.90i·10-s + (−1.14 + 0.658i)11-s + (−0.838 + 2.16i)12-s + (−0.317 − 0.183i)13-s + (−0.376 + 0.972i)15-s + (−1.03 − 1.79i)16-s + 1.40·17-s + (1.22 − 1.34i)18-s − 0.154i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.445 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.445 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.445 + 0.895i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.445 + 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34757 - 2.17604i\)
\(L(\frac12)\) \(\approx\) \(1.34757 - 2.17604i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.71 - 0.266i)T \)
7 \( 1 \)
good2 \( 1 + (-2.23 + 1.28i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-1.16 + 2.01i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.78 - 2.18i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.14 + 0.660i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 5.78T + 17T^{2} \)
19 \( 1 + 0.675iT - 19T^{2} \)
23 \( 1 + (4.81 + 2.78i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.86 + 2.23i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.47 - 2.00i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 3.01T + 37T^{2} \)
41 \( 1 + (3.29 - 5.70i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.89 - 6.74i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.246 - 0.427i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 4.14iT - 53T^{2} \)
59 \( 1 + (-2.15 + 3.73i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.77 + 1.02i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.41 + 4.17i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.17iT - 71T^{2} \)
73 \( 1 - 15.1iT - 73T^{2} \)
79 \( 1 + (-5.30 - 9.18i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.32 + 9.22i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 3.32T + 89T^{2} \)
97 \( 1 + (12.7 - 7.36i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.07597490529393829446351418528, −10.04978356905516247230602134541, −9.857023397235372417540310568116, −7.908553262161534569934209762488, −6.48055602605207676235607635316, −5.53828556833233252112026620285, −5.02469024875018490878154169576, −4.23620199662155799080327620779, −2.67745553760474078730344754395, −1.23637704993691545606675173715, 2.54988147102074352121965595665, 3.74402770911954257143266859684, 5.09084371857227739326564199741, 5.71472314450043749682287316874, 6.39110227018934910123066720793, 7.32462716030017463586026754146, 8.023309809616399556135218564536, 10.04322912168750928924886493296, 10.69316583843405378524805131680, 11.81958361289397601686709012580

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