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Properties

Label 2-21e2-63.25-c1-0-3
Degree $2$
Conductor $441$
Sign $-0.998 - 0.0530i$
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  − 1.72·2-s + (−1.70 + 0.276i)3-s + 0.981·4-s + (−1.75 + 3.04i)5-s + (2.95 − 0.477i)6-s + 1.75·8-s + (2.84 − 0.946i)9-s + (3.03 − 5.25i)10-s + (3.04 + 5.27i)11-s + (−1.67 + 0.271i)12-s + (0.560 + 0.970i)13-s + (2.16 − 5.68i)15-s − 4.99·16-s + (−0.601 + 1.04i)17-s + (−4.91 + 1.63i)18-s + (1.10 + 1.90i)19-s + ⋯
L(s)  = 1  − 1.22·2-s + (−0.987 + 0.159i)3-s + 0.490·4-s + (−0.785 + 1.36i)5-s + (1.20 − 0.195i)6-s + 0.621·8-s + (0.948 − 0.315i)9-s + (0.958 − 1.66i)10-s + (0.918 + 1.59i)11-s + (−0.484 + 0.0783i)12-s + (0.155 + 0.269i)13-s + (0.557 − 1.46i)15-s − 1.24·16-s + (−0.146 + 0.252i)17-s + (−1.15 + 0.385i)18-s + (0.252 + 0.438i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0530i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0530i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.998 - 0.0530i$
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: \((2,\ 441,\ (\ :1/2),\ -0.998 - 0.0530i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00786258 + 0.296404i\)
\(L(\frac12)\) \(\approx\) \(0.00786258 + 0.296404i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.70 - 0.276i)T \)
7 \( 1 \)
good2 \( 1 + 1.72T + 2T^{2} \)
5 \( 1 + (1.75 - 3.04i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.04 - 5.27i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.560 - 0.970i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.601 - 1.04i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.10 - 1.90i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.636 + 1.10i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.10 - 5.37i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.188T + 31T^{2} \)
37 \( 1 + (1.78 + 3.09i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.68 - 2.91i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.90 - 3.29i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.72T + 47T^{2} \)
53 \( 1 + (-4.16 + 7.22i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 11.2T + 59T^{2} \)
61 \( 1 - 12.0T + 61T^{2} \)
67 \( 1 + 7.91T + 67T^{2} \)
71 \( 1 + 12.2T + 71T^{2} \)
73 \( 1 + (-2.65 + 4.60i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 9.21T + 79T^{2} \)
83 \( 1 + (-0.624 + 1.08i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.77 + 4.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.24 + 14.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.32008537612642749139298615868, −10.53918167447863930147982020923, −9.943734045770906235359050308907, −9.080897034968135721029770137534, −7.68113735549747144151456512923, −7.08155066809315242464773925809, −6.44350283441723140239571031205, −4.72253707595297094584414087898, −3.75462655913305783626440773941, −1.70756994147431821787683358470, 0.38347389778777030423567470737, 1.24334862101216035511539138570, 3.91240072540950555943020405836, 4.92572719173772600231667259158, 5.99580590688764171128671063395, 7.24113016964582911459555112711, 8.160431262925640721546996044198, 8.836205747395950255631741273520, 9.573501629201345873375558032508, 10.77518563985246622840527510726

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