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: $\chi_{441} (295, \cdot )$
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

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

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