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Properties

Label 2-21e2-63.41-c1-0-16
Degree $2$
Conductor $441$
Sign $0.844 + 0.536i$
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  + (−2.34 + 1.35i)2-s + (−1.70 − 0.322i)3-s + (2.66 − 4.62i)4-s + (0.601 − 1.04i)5-s + (4.42 − 1.54i)6-s + 9.04i·8-s + (2.79 + 1.09i)9-s + 3.25i·10-s + (−2.15 + 1.24i)11-s + (−6.03 + 7.00i)12-s + (1.63 + 0.942i)13-s + (−1.35 + 1.57i)15-s + (−6.90 − 11.9i)16-s − 1.20·17-s + (−8.03 + 1.21i)18-s − 7.47i·19-s + ⋯
L(s)  = 1  + (−1.65 + 0.957i)2-s + (−0.982 − 0.185i)3-s + (1.33 − 2.31i)4-s + (0.268 − 0.465i)5-s + (1.80 − 0.632i)6-s + 3.19i·8-s + (0.930 + 0.365i)9-s + 1.03i·10-s + (−0.649 + 0.374i)11-s + (−1.74 + 2.02i)12-s + (0.452 + 0.261i)13-s + (−0.350 + 0.407i)15-s + (−1.72 − 2.99i)16-s − 0.291·17-s + (−1.89 + 0.285i)18-s − 1.71i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 + 0.536i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.844 + 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.844 + 0.536i$
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: \((2,\ 441,\ (\ :1/2),\ 0.844 + 0.536i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.373727 - 0.108660i\)
\(L(\frac12)\) \(\approx\) \(0.373727 - 0.108660i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.70 + 0.322i)T \)
7 \( 1 \)
good2 \( 1 + (2.34 - 1.35i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-0.601 + 1.04i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.15 - 1.24i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.63 - 0.942i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.20T + 17T^{2} \)
19 \( 1 + 7.47iT - 19T^{2} \)
23 \( 1 + (-2.63 - 1.52i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.173 - 0.100i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.03 + 1.75i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.73T + 37T^{2} \)
41 \( 1 + (-3.36 + 5.82i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.00656 - 0.0113i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.717 + 1.24i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 9.90iT - 53T^{2} \)
59 \( 1 + (-6.10 + 10.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9.73 + 5.62i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.57 + 4.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.0iT - 71T^{2} \)
73 \( 1 - 8.67iT - 73T^{2} \)
79 \( 1 + (2.74 + 4.75i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.60 - 2.78i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 7.96T + 89T^{2} \)
97 \( 1 + (-2.06 + 1.19i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.10723423819098387958304600472, −9.877106520901964223818737413254, −9.245325169644655620954495620403, −8.324799402534247780517149135050, −7.21011289215138578436466518251, −6.73945862095338599802231743040, −5.58290606383996376564468076346, −4.92129572706625752118042973588, −1.97297047329497572512812257473, −0.55877843328159735071183511968, 1.16724653519384935146518526620, 2.68611729077578959780373498997, 3.96552865524233356321607029896, 5.78199459968213022182995010845, 6.78582118683484232502708379147, 7.77424026258175511584497916345, 8.664468006323119660814635302125, 9.752635440414407685155145002026, 10.44766294017533094163375560117, 10.83514567370914410554985212546

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