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Properties

Label 2-21e2-9.4-c1-0-15
Degree $2$
Conductor $441$
Sign $0.145 - 0.989i$
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  + (0.863 + 1.49i)2-s + (1.09 − 1.34i)3-s + (−0.490 + 0.849i)4-s + (−1.75 + 3.04i)5-s + (2.95 + 0.477i)6-s + 1.75·8-s + (−0.604 − 2.93i)9-s − 6.06·10-s + (3.04 + 5.27i)11-s + (0.603 + 1.58i)12-s + (0.560 − 0.970i)13-s + (2.16 + 5.68i)15-s + (2.49 + 4.32i)16-s + 1.20·17-s + (3.87 − 3.44i)18-s − 2.20·19-s + ⋯
L(s)  = 1  + (0.610 + 1.05i)2-s + (0.631 − 0.775i)3-s + (−0.245 + 0.424i)4-s + (−0.785 + 1.36i)5-s + (1.20 + 0.195i)6-s + 0.621·8-s + (−0.201 − 0.979i)9-s − 1.91·10-s + (0.918 + 1.59i)11-s + (0.174 + 0.458i)12-s + (0.155 − 0.269i)13-s + (0.557 + 1.46i)15-s + (0.624 + 1.08i)16-s + 0.292·17-s + (0.912 − 0.810i)18-s − 0.505·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.145 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.145 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.145 - 0.989i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.145 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69493 + 1.46342i\)
\(L(\frac12)\) \(\approx\) \(1.69493 + 1.46342i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.09 + 1.34i)T \)
7 \( 1 \)
good2 \( 1 + (-0.863 - 1.49i)T + (-1 + 1.73i)T^{2} \)
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 \( 1 - 1.20T + 17T^{2} \)
19 \( 1 + 2.20T + 19T^{2} \)
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 \( 1 + (-0.0942 + 0.163i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.57T + 37T^{2} \)
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 \( 1 + (-2.86 - 4.95i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 8.33T + 53T^{2} \)
59 \( 1 + (-5.63 + 9.75i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.00 + 10.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.95 + 6.85i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.2T + 71T^{2} \)
73 \( 1 + 5.31T + 73T^{2} \)
79 \( 1 + (4.60 + 7.98i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.624 - 1.08i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 5.54T + 89T^{2} \)
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

−11.49966189880276080462164163792, −10.45510194780656186696350744877, −9.388343563394410937853008241058, −7.953542239201088021583637894423, −7.47034981495467440035739626526, −6.73749822658255689256148504269, −6.16436379372828191641691737830, −4.45418744382535952177722669396, −3.52762856352551706027146492685, −2.04530053404844472302346584818, 1.31527148185784062019939151292, 3.09784807777428752022436824017, 3.91672858398396948013643330664, 4.56914159435293335202407029520, 5.67817352912794795834190800573, 7.55955347667867912317761952799, 8.604756488671058171328518102074, 8.969717939693909810815870188928, 10.18006186122097308735033770226, 11.29302684687501420047948544433

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