Properties

Label 2-21e2-441.142-c1-0-6
Degree $2$
Conductor $441$
Sign $-0.497 - 0.867i$
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  + (−0.0677 + 0.172i)2-s + (0.479 − 1.66i)3-s + (1.44 + 1.33i)4-s + (−3.85 + 1.85i)5-s + (0.254 + 0.195i)6-s + (−2.13 − 1.56i)7-s + (−0.662 + 0.318i)8-s + (−2.54 − 1.59i)9-s + (−0.0593 − 0.791i)10-s + (2.63 + 3.30i)11-s + (2.91 − 1.75i)12-s + (−1.78 + 4.55i)13-s + (0.414 − 0.262i)14-s + (1.24 + 7.31i)15-s + (0.283 + 3.78i)16-s + (−3.42 + 3.17i)17-s + ⋯
L(s)  = 1  + (−0.0478 + 0.121i)2-s + (0.276 − 0.960i)3-s + (0.720 + 0.668i)4-s + (−1.72 + 0.831i)5-s + (0.103 + 0.0797i)6-s + (−0.806 − 0.591i)7-s + (−0.234 + 0.112i)8-s + (−0.846 − 0.532i)9-s + (−0.0187 − 0.250i)10-s + (0.795 + 0.997i)11-s + (0.841 − 0.507i)12-s + (−0.495 + 1.26i)13-s + (0.110 − 0.0700i)14-s + (0.320 + 1.88i)15-s + (0.0709 + 0.946i)16-s + (−0.830 + 0.770i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.497 - 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.497 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.497 - 0.867i$
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: \((2,\ 441,\ (\ :1/2),\ -0.497 - 0.867i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.334901 + 0.578247i\)
\(L(\frac12)\) \(\approx\) \(0.334901 + 0.578247i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.479 + 1.66i)T \)
7 \( 1 + (2.13 + 1.56i)T \)
good2 \( 1 + (0.0677 - 0.172i)T + (-1.46 - 1.36i)T^{2} \)
5 \( 1 + (3.85 - 1.85i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (-2.63 - 3.30i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (1.78 - 4.55i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (3.42 - 3.17i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (-1.76 + 3.05i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.991 - 4.34i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (4.35 + 1.34i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (0.399 - 0.691i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.39 + 0.428i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (5.38 + 3.66i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (-0.803 + 0.547i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (0.527 - 1.34i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (-3.59 + 1.10i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (-7.83 + 5.34i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (-3.42 + 3.18i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (-1.97 + 3.42i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.472 - 2.07i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (6.12 + 0.923i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (-7.70 - 13.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.583 - 1.48i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (-1.85 - 4.73i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (3.41 - 5.90i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.61090957836356649513472966328, −10.95277534939511041609847584879, −9.450952909282233888241321233025, −8.367690816522696070668634282341, −7.36463428026789325922267863018, −6.98878672800267692418207955534, −6.57731307958533518995907573399, −4.08554577402935039601839802576, −3.49597492117681469069591905488, −2.17283181004329395025431339915, 0.38744735143121769907276900986, 2.91220724871658262028564369009, 3.71913548093883462465473762276, 4.98085539581687105044459422681, 5.87446430968405899943097609123, 7.22231456254759332784847632863, 8.350979508536837236280185994434, 8.977132574488139896371169736943, 9.934707892241496364440271158697, 10.90706575374562156267470621190

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