Properties

Degree $2$
Conductor $441$
Sign $-0.533 + 0.845i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.19 + 2.06i)2-s + (−1.34 + 1.08i)3-s + (−1.84 + 3.20i)4-s + (−1.46 + 2.52i)5-s + (−3.85 − 1.49i)6-s − 4.05·8-s + (0.637 − 2.93i)9-s − 6.97·10-s + (0.676 + 1.17i)11-s + (−0.987 − 6.32i)12-s + (0.733 − 1.26i)13-s + (−0.779 − 4.99i)15-s + (−1.13 − 1.96i)16-s + 3.31·17-s + (6.82 − 2.18i)18-s − 2.20·19-s + ⋯
L(s)  = 1  + (0.843 + 1.46i)2-s + (−0.778 + 0.627i)3-s + (−0.924 + 1.60i)4-s + (−0.653 + 1.13i)5-s + (−1.57 − 0.608i)6-s − 1.43·8-s + (0.212 − 0.977i)9-s − 2.20·10-s + (0.204 + 0.353i)11-s + (−0.284 − 1.82i)12-s + (0.203 − 0.352i)13-s + (−0.201 − 1.29i)15-s + (−0.284 − 0.492i)16-s + 0.802·17-s + (1.60 − 0.514i)18-s − 0.506·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.533 + 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.533 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.533 + 0.845i$
Motivic weight: \(1\)
Character: $\chi_{441} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.533 + 0.845i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.602978 - 1.09375i\)
\(L(\frac12)\) \(\approx\) \(0.602978 - 1.09375i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.34 - 1.08i)T \)
7 \( 1 \)
good2 \( 1 + (-1.19 - 2.06i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.46 - 2.52i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.676 - 1.17i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.733 + 1.26i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.31T + 17T^{2} \)
19 \( 1 + 2.20T + 19T^{2} \)
23 \( 1 + (1.31 - 2.27i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.521 - 0.903i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.63 + 2.83i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 10.8T + 37T^{2} \)
41 \( 1 + (-0.904 + 1.56i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.17 + 3.76i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.98 - 3.44i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6.45T + 53T^{2} \)
59 \( 1 + (6.10 - 10.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.279 - 0.484i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.40 - 11.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.9T + 71T^{2} \)
73 \( 1 - 10.4T + 73T^{2} \)
79 \( 1 + (0.383 + 0.664i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.983 - 1.70i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 6.40T + 89T^{2} \)
97 \( 1 + (-4.14 - 7.17i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.87055314109708894295715350268, −10.82590794373649655778197842199, −10.08711330824915765629522928689, −8.715155576009834591145585379830, −7.55928098599017941844376470604, −6.93151182948344019294045198062, −6.07401024717069481969169658391, −5.23998408679832331047864623766, −4.10096284721631459774211876902, −3.37831316458771277531297714115, 0.69268386286219370261094706363, 1.86509311907150837187415458838, 3.54152298639352697267958007160, 4.61525112765182431446918877846, 5.27413228531455941169837461247, 6.44326981167731814017139311174, 7.88382451393826660369771637269, 8.832245708751951561358042530727, 10.08268629103618180451313034379, 10.90976832414913408482272454931

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