Properties

Label 2-21e2-63.25-c1-0-2
Degree $2$
Conductor $441$
Sign $-0.743 + 0.668i$
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  − 0.495·2-s + (0.221 + 1.71i)3-s − 1.75·4-s + (−1.84 + 3.19i)5-s + (−0.109 − 0.851i)6-s + 1.86·8-s + (−2.90 + 0.760i)9-s + (0.915 − 1.58i)10-s + (0.446 + 0.772i)11-s + (−0.388 − 3.01i)12-s + (−0.598 − 1.03i)13-s + (−5.90 − 2.46i)15-s + 2.58·16-s + (0.124 − 0.216i)17-s + (1.43 − 0.377i)18-s + (−1.40 − 2.43i)19-s + ⋯
L(s)  = 1  − 0.350·2-s + (0.127 + 0.991i)3-s − 0.877·4-s + (−0.825 + 1.43i)5-s + (−0.0447 − 0.347i)6-s + 0.658·8-s + (−0.967 + 0.253i)9-s + (0.289 − 0.501i)10-s + (0.134 + 0.233i)11-s + (−0.112 − 0.869i)12-s + (−0.165 − 0.287i)13-s + (−1.52 − 0.636i)15-s + 0.646·16-s + (0.0303 − 0.0525i)17-s + (0.339 − 0.0888i)18-s + (−0.322 − 0.557i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.743 + 0.668i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.743 + 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.743 + 0.668i$
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: \((2,\ 441,\ (\ :1/2),\ -0.743 + 0.668i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.121295 - 0.316453i\)
\(L(\frac12)\) \(\approx\) \(0.121295 - 0.316453i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.221 - 1.71i)T \)
7 \( 1 \)
good2 \( 1 + 0.495T + 2T^{2} \)
5 \( 1 + (1.84 - 3.19i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.446 - 0.772i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.598 + 1.03i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.124 + 0.216i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.40 + 2.43i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.23 - 2.14i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.07 + 3.58i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.58T + 31T^{2} \)
37 \( 1 + (2.36 + 4.09i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.39 - 4.14i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.98 - 8.64i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 10.1T + 47T^{2} \)
53 \( 1 + (4.94 - 8.56i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 1.81T + 59T^{2} \)
61 \( 1 + 10.8T + 61T^{2} \)
67 \( 1 - 1.02T + 67T^{2} \)
71 \( 1 + 4.94T + 71T^{2} \)
73 \( 1 + (-0.915 + 1.58i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 1.79T + 79T^{2} \)
83 \( 1 + (6.16 - 10.6i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.20 - 2.08i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.52 - 9.56i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.25982691083139035313651661295, −10.69395513952759450597185011873, −9.908178791560010448569318222385, −9.131251270310413870669918904021, −8.076853539609268527437507566838, −7.35197452738559865138993124101, −5.99334932136440708330804519991, −4.65754597820082198942485249991, −3.84534710279292482956267870239, −2.83794593389116415653035430926, 0.25186025295442320195727123095, 1.53943768469215698140188523372, 3.66284566054068512990272851749, 4.71552361151597718266871539046, 5.70393856883144202334991756743, 7.16050635247988401338053380957, 8.086835298019476903374865798190, 8.643278510269076242490260163630, 9.208111674343385174985101477119, 10.53209592860531972189902191149

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