Properties

Label 2-21e2-63.25-c1-0-18
Degree $2$
Conductor $441$
Sign $0.172 - 0.985i$
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  + 2.17·2-s + (−0.507 + 1.65i)3-s + 2.73·4-s + (−0.634 + 1.09i)5-s + (−1.10 + 3.60i)6-s + 1.60·8-s + (−2.48 − 1.68i)9-s + (−1.38 + 2.39i)10-s + (2.73 + 4.74i)11-s + (−1.38 + 4.53i)12-s + (2.37 + 4.10i)13-s + (−1.49 − 1.60i)15-s − 1.98·16-s + (2.40 − 4.17i)17-s + (−5.40 − 3.65i)18-s + (−2.69 − 4.66i)19-s + ⋯
L(s)  = 1  + 1.53·2-s + (−0.292 + 0.956i)3-s + 1.36·4-s + (−0.283 + 0.491i)5-s + (−0.450 + 1.47i)6-s + 0.566·8-s + (−0.828 − 0.560i)9-s + (−0.436 + 0.755i)10-s + (0.825 + 1.43i)11-s + (−0.400 + 1.30i)12-s + (0.658 + 1.13i)13-s + (−0.386 − 0.415i)15-s − 0.496·16-s + (0.584 − 1.01i)17-s + (−1.27 − 0.862i)18-s + (−0.617 − 1.06i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.172 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.172 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.172 - 0.985i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (214, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.172 - 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.05892 + 1.73034i\)
\(L(\frac12)\) \(\approx\) \(2.05892 + 1.73034i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.507 - 1.65i)T \)
7 \( 1 \)
good2 \( 1 - 2.17T + 2T^{2} \)
5 \( 1 + (0.634 - 1.09i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.73 - 4.74i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.37 - 4.10i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.40 + 4.17i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.69 + 4.66i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.58 + 4.48i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.01 + 3.49i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.46T + 31T^{2} \)
37 \( 1 + (0.959 + 1.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.94 + 3.37i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.66 - 2.87i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.15T + 47T^{2} \)
53 \( 1 + (-3.57 + 6.18i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 0.308T + 59T^{2} \)
61 \( 1 - 10.3T + 61T^{2} \)
67 \( 1 - 4.47T + 67T^{2} \)
71 \( 1 + 1.96T + 71T^{2} \)
73 \( 1 + (-5.27 + 9.13i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 9.01T + 79T^{2} \)
83 \( 1 + (5.08 - 8.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.59 + 4.49i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.48 - 4.30i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.55362691368634136517285298492, −10.80235402861382010973348976474, −9.597511896299537830399438592211, −8.876175524062463336959245599633, −6.94925592972985382430555178461, −6.60105622812370260092029235409, −5.21757980670305588106533156148, −4.45257132792567769201424238935, −3.77653935612226070298225060344, −2.53632542595694746778699367058, 1.24840337884961947607914256880, 3.10707033446095975732885003467, 3.92176589168283536834415631832, 5.47031277871544508005618468531, 5.87318436795948174074678909652, 6.77214006050267005452613493245, 8.157092238155118479961596449857, 8.657144747550651384261073732364, 10.54971740164793884499380919946, 11.34011839086354931678261862628

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