Properties

Label 2-21e2-9.7-c1-0-27
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: $\chi_{441} (295, \cdot )$
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

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

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