Properties

Label 2-21e2-441.142-c1-0-4
Degree $2$
Conductor $441$
Sign $0.940 - 0.338i$
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  + (0.969 − 2.47i)2-s + (−1.62 + 0.596i)3-s + (−3.70 − 3.43i)4-s + (−2.70 + 1.30i)5-s + (−0.102 + 4.59i)6-s + (2.23 + 1.42i)7-s + (−7.29 + 3.51i)8-s + (2.28 − 1.94i)9-s + (0.595 + 7.94i)10-s + (0.141 + 0.177i)11-s + (8.06 + 3.37i)12-s + (−1.85 + 4.73i)13-s + (5.67 − 4.13i)14-s + (3.62 − 3.73i)15-s + (0.851 + 11.3i)16-s + (−3.43 + 3.18i)17-s + ⋯
L(s)  = 1  + (0.685 − 1.74i)2-s + (−0.938 + 0.344i)3-s + (−1.85 − 1.71i)4-s + (−1.20 + 0.582i)5-s + (−0.0417 + 1.87i)6-s + (0.843 + 0.537i)7-s + (−2.57 + 1.24i)8-s + (0.762 − 0.646i)9-s + (0.188 + 2.51i)10-s + (0.0426 + 0.0534i)11-s + (2.32 + 0.974i)12-s + (−0.515 + 1.31i)13-s + (1.51 − 1.10i)14-s + (0.934 − 0.963i)15-s + (0.212 + 2.84i)16-s + (−0.832 + 0.772i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.338i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.940 - 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.940 - 0.338i$
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: \((2,\ 441,\ (\ :1/2),\ 0.940 - 0.338i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.555974 + 0.0969977i\)
\(L(\frac12)\) \(\approx\) \(0.555974 + 0.0969977i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.62 - 0.596i)T \)
7 \( 1 + (-2.23 - 1.42i)T \)
good2 \( 1 + (-0.969 + 2.47i)T + (-1.46 - 1.36i)T^{2} \)
5 \( 1 + (2.70 - 1.30i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (-0.141 - 0.177i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (1.85 - 4.73i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (3.43 - 3.18i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (-2.97 + 5.15i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.24 - 5.45i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (0.773 + 0.238i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (-0.361 + 0.626i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.12 - 0.347i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (-1.34 - 0.916i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (6.33 - 4.31i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (2.96 - 7.54i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (6.19 - 1.91i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (11.0 - 7.54i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (-0.404 + 0.375i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (1.24 - 2.16i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.70 + 11.8i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (9.33 + 1.40i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (-6.26 - 10.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.79 + 9.66i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (1.29 + 3.30i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (-7.78 + 13.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.34480327206161701219361419605, −10.92422624230259034772487007300, −9.719461920957556919271726333129, −8.963375679969110648085563428213, −7.42613471766168791487811619561, −6.08768961933636102564574985876, −4.73116722215672373916213055314, −4.40329502947387111695904705078, −3.21165926512184539647375291070, −1.69814299244354340020945411515, 0.33194444323819209242307872279, 3.78330038531747851603859299938, 4.78332383039105783631140207528, 5.19700060634155184754314441434, 6.40930631510139770696442546727, 7.45891765040509736595047698718, 7.82725774872884422043193319313, 8.577153739667277104493298898641, 10.20133053290630964028905999187, 11.44958705041759773698315499850

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