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Properties

Label 2-21e2-63.5-c1-0-22
Degree $2$
Conductor $441$
Sign $0.746 + 0.665i$
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  − 0.981i·2-s + (1.73 + 0.0395i)3-s + 1.03·4-s + (0.940 + 1.62i)5-s + (0.0387 − 1.69i)6-s − 2.98i·8-s + (2.99 + 0.136i)9-s + (1.59 − 0.923i)10-s + (−3.54 − 2.04i)11-s + (1.79 + 0.0409i)12-s + (−3.51 − 2.02i)13-s + (1.56 + 2.85i)15-s − 0.852·16-s + (0.810 + 1.40i)17-s + (0.134 − 2.94i)18-s + (7.03 + 4.06i)19-s + ⋯
L(s)  = 1  − 0.694i·2-s + (0.999 + 0.0228i)3-s + 0.518·4-s + (0.420 + 0.728i)5-s + (0.0158 − 0.693i)6-s − 1.05i·8-s + (0.998 + 0.0456i)9-s + (0.505 − 0.291i)10-s + (−1.06 − 0.616i)11-s + (0.518 + 0.0118i)12-s + (−0.974 − 0.562i)13-s + (0.403 + 0.737i)15-s − 0.213·16-s + (0.196 + 0.340i)17-s + (0.0316 − 0.693i)18-s + (1.61 + 0.932i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 + 0.665i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.746 + 0.665i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.746 + 0.665i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.13980 - 0.816022i\)
\(L(\frac12)\) \(\approx\) \(2.13980 - 0.816022i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.73 - 0.0395i)T \)
7 \( 1 \)
good2 \( 1 + 0.981iT - 2T^{2} \)
5 \( 1 + (-0.940 - 1.62i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.54 + 2.04i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.51 + 2.02i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.810 - 1.40i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-7.03 - 4.06i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.73 - 2.15i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.542 - 0.313i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.27iT - 31T^{2} \)
37 \( 1 + (3.97 - 6.87i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.912 - 1.57i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.53 + 6.12i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 7.93T + 47T^{2} \)
53 \( 1 + (7.24 - 4.18i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 8.17T + 59T^{2} \)
61 \( 1 + 3.74iT - 61T^{2} \)
67 \( 1 - 12.5T + 67T^{2} \)
71 \( 1 + 14.4iT - 71T^{2} \)
73 \( 1 + (3.28 - 1.89i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 8.36T + 79T^{2} \)
83 \( 1 + (4.38 + 7.59i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.90 - 8.49i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (11.4 - 6.61i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.74365004537041989793479811810, −10.12582720860950509015571307300, −9.668929218217843387661237302037, −8.115167481363932474437210669187, −7.53997284012776768965264165445, −6.47884067651378821034096379112, −5.20314731666376681397347722542, −3.37028670931072517211114598591, −2.94860416863773417904036337145, −1.74976736412818087423305288218, 1.92131013193701217666184026350, 2.89555702541331205230383119790, 4.71367636475564449005039094704, 5.41247175992671225443265160161, 6.91922010176402365605594872766, 7.52061774165697902799114300865, 8.291235066953149086762971128959, 9.436146382549766800402781661190, 9.899674352990749363187280719337, 11.25270472193120654066119947316

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