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Properties

Label 2-21e2-63.20-c1-0-29
Degree $2$
Conductor $441$
Sign $-0.995 + 0.0896i$
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  + (0.575 + 0.332i)2-s + (−1.69 − 0.358i)3-s + (−0.779 − 1.34i)4-s + (0.0141 + 0.0245i)5-s + (−0.855 − 0.768i)6-s − 2.36i·8-s + (2.74 + 1.21i)9-s + 0.0188i·10-s + (−0.885 − 0.511i)11-s + (0.837 + 2.56i)12-s + (−4.87 + 2.81i)13-s + (−0.0152 − 0.0466i)15-s + (−0.773 + 1.33i)16-s − 5.67·17-s + (1.17 + 1.60i)18-s − 2.09i·19-s + ⋯
L(s)  = 1  + (0.406 + 0.234i)2-s + (−0.978 − 0.206i)3-s + (−0.389 − 0.674i)4-s + (0.00632 + 0.0109i)5-s + (−0.349 − 0.313i)6-s − 0.835i·8-s + (0.914 + 0.404i)9-s + 0.00594i·10-s + (−0.266 − 0.154i)11-s + (0.241 + 0.740i)12-s + (−1.35 + 0.781i)13-s + (−0.00392 − 0.0120i)15-s + (−0.193 + 0.334i)16-s − 1.37·17-s + (0.276 + 0.379i)18-s − 0.480i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.995 + 0.0896i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.995 + 0.0896i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0102740 - 0.228860i\)
\(L(\frac12)\) \(\approx\) \(0.0102740 - 0.228860i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.69 + 0.358i)T \)
7 \( 1 \)
good2 \( 1 + (-0.575 - 0.332i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-0.0141 - 0.0245i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.885 + 0.511i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.87 - 2.81i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.67T + 17T^{2} \)
19 \( 1 + 2.09iT - 19T^{2} \)
23 \( 1 + (6.28 - 3.63i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.52 + 2.03i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.87 + 1.65i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.47T + 37T^{2} \)
41 \( 1 + (-3.52 - 6.11i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.15 - 2.00i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.43 + 9.42i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 11.5iT - 53T^{2} \)
59 \( 1 + (3.01 + 5.21i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.05 - 1.18i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.38 + 11.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.93iT - 71T^{2} \)
73 \( 1 + 10.8iT - 73T^{2} \)
79 \( 1 + (-7.80 + 13.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.07 + 5.32i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 12.0T + 89T^{2} \)
97 \( 1 + (-6.77 - 3.91i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.69716055412894973240049607420, −9.906853739222246792155802052902, −9.107460772572520875656429393250, −7.60641870491479147630670153931, −6.66073178760268822464257821655, −5.95176703094155053862045158413, −4.84278735833021257717291958584, −4.29850458476672199736508212879, −2.06437189095639192997637523371, −0.13500862193164852903799118810, 2.41817860143156332107597745180, 3.91091353441460865686600767611, 4.80305011401930076700035034202, 5.58836342996261966614388459747, 6.91451409130666598005136508048, 7.78284633717918029695478540192, 8.921472757605479039855648552902, 9.969621152277646513436050758737, 10.78980528238671306357721927695, 11.67202775249802766125638539710

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