Properties

Label 2-21e2-63.58-c1-0-9
Degree $2$
Conductor $441$
Sign $0.909 + 0.416i$
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  − 1.10·2-s + (−1.22 − 1.22i)3-s − 0.783·4-s + (−0.0527 − 0.0913i)5-s + (1.34 + 1.35i)6-s + 3.07·8-s + (−0.0231 + 2.99i)9-s + (0.0581 + 0.100i)10-s + (−1.66 + 2.89i)11-s + (0.956 + 0.963i)12-s + (−1.23 + 2.14i)13-s + (−0.0479 + 0.176i)15-s − 1.81·16-s + (−0.806 − 1.39i)17-s + (0.0255 − 3.30i)18-s + (3.84 − 6.65i)19-s + ⋯
L(s)  = 1  − 0.779·2-s + (−0.704 − 0.709i)3-s − 0.391·4-s + (−0.0235 − 0.0408i)5-s + (0.549 + 0.553i)6-s + 1.08·8-s + (−0.00772 + 0.999i)9-s + (0.0183 + 0.0318i)10-s + (−0.503 + 0.871i)11-s + (0.276 + 0.278i)12-s + (−0.343 + 0.595i)13-s + (−0.0123 + 0.0455i)15-s − 0.454·16-s + (−0.195 − 0.338i)17-s + (0.00602 − 0.779i)18-s + (0.881 − 1.52i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 + 0.416i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.909 + 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.909 + 0.416i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.909 + 0.416i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.570843 - 0.124483i\)
\(L(\frac12)\) \(\approx\) \(0.570843 - 0.124483i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.22 + 1.22i)T \)
7 \( 1 \)
good2 \( 1 + 1.10T + 2T^{2} \)
5 \( 1 + (0.0527 + 0.0913i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.66 - 2.89i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.23 - 2.14i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.806 + 1.39i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.84 + 6.65i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.948 - 1.64i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.64 - 8.04i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 9.26T + 31T^{2} \)
37 \( 1 + (-0.991 + 1.71i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.74 + 6.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.77 + 6.53i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.19T + 47T^{2} \)
53 \( 1 + (-4.98 - 8.64i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 4.45T + 59T^{2} \)
61 \( 1 + 5.67T + 61T^{2} \)
67 \( 1 - 9.97T + 67T^{2} \)
71 \( 1 - 3.29T + 71T^{2} \)
73 \( 1 + (-2.36 - 4.09i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 7.69T + 79T^{2} \)
83 \( 1 + (0.584 + 1.01i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.01 - 5.22i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.90 + 3.29i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.91891472365187781948794973891, −10.17060573473639250144316638505, −9.243461209518037518191865863374, −8.343807401125866400035774957580, −7.27196519114016874069842740293, −6.82082746241027914354828788585, −5.14508747804976273133422060479, −4.62377071645734803476064546134, −2.43329831352209337251276493718, −0.855388311194051926557906523150, 0.848367774066355131763228215115, 3.22759772254401739002411378053, 4.48605562568878583029367141046, 5.41316537396468239912058948354, 6.40470953747152257728877014268, 7.912815615899584683511210927164, 8.420585959026046866200970292632, 9.765089887931507504054176409743, 10.01267515516036057802196235067, 10.95015392843709681250339669587

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