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Properties

Label 2-21e2-63.41-c1-0-5
Degree $2$
Conductor $441$
Sign $-0.997 + 0.0754i$
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  + (−1.02 + 0.589i)2-s + (1.61 + 0.613i)3-s + (−0.305 + 0.529i)4-s + (−2.16 + 3.75i)5-s + (−2.01 + 0.327i)6-s − 3.07i·8-s + (2.24 + 1.98i)9-s − 5.10i·10-s + (−1.87 + 1.08i)11-s + (−0.820 + 0.670i)12-s + (2.25 + 1.30i)13-s + (−5.81 + 4.74i)15-s + (1.20 + 2.08i)16-s − 1.17·17-s + (−3.46 − 0.706i)18-s − 2.41i·19-s + ⋯
L(s)  = 1  + (−0.721 + 0.416i)2-s + (0.935 + 0.354i)3-s + (−0.152 + 0.264i)4-s + (−0.968 + 1.67i)5-s + (−0.822 + 0.133i)6-s − 1.08i·8-s + (0.748 + 0.662i)9-s − 1.61i·10-s + (−0.564 + 0.325i)11-s + (−0.236 + 0.193i)12-s + (0.624 + 0.360i)13-s + (−1.50 + 1.22i)15-s + (0.300 + 0.520i)16-s − 0.284·17-s + (−0.816 − 0.166i)18-s − 0.554i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.997 + 0.0754i$
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: \((2,\ 441,\ (\ :1/2),\ -0.997 + 0.0754i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0307666 - 0.814361i\)
\(L(\frac12)\) \(\approx\) \(0.0307666 - 0.814361i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.61 - 0.613i)T \)
7 \( 1 \)
good2 \( 1 + (1.02 - 0.589i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (2.16 - 3.75i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.87 - 1.08i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.25 - 1.30i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.17T + 17T^{2} \)
19 \( 1 + 2.41iT - 19T^{2} \)
23 \( 1 + (3.16 + 1.82i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.589 + 0.340i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.67 - 3.27i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 5.10T + 37T^{2} \)
41 \( 1 + (3.68 - 6.38i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.12 + 3.68i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.57 - 6.18i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 3.23iT - 53T^{2} \)
59 \( 1 + (-2.91 + 5.05i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.21 + 3.58i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.32 - 5.76i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.95iT - 71T^{2} \)
73 \( 1 - 11.9iT - 73T^{2} \)
79 \( 1 + (-4.87 - 8.44i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.796 - 1.37i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 6.09T + 89T^{2} \)
97 \( 1 + (2.36 - 1.36i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.29146416479195363793702605391, −10.41253264247122104241830189791, −9.809852720647400952286358756284, −8.556465932930279397153732406243, −8.061004939216696017757541592921, −7.15794630548048673048204415630, −6.60701265784022945652584335727, −4.41031639371294793624974511325, −3.55334707472889891722800436091, −2.62162240104953961908345266255, 0.59796395363790213566909929087, 1.81056531720636872691329524610, 3.56528279180644407694825376157, 4.67352999322547617348140279708, 5.76902803570710381793908629736, 7.55757042336670852580659821888, 8.403556817395161926650429566535, 8.571516543682199218843555608163, 9.541882064220432937700928307181, 10.44168585530615980841195179441

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