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Properties

Label 2-21e2-21.5-c3-0-19
Degree $2$
Conductor $441$
Sign $-0.932 - 0.360i$
Analytic cond. $26.0198$
Root an. cond. $5.10096$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.68 + 2.70i)2-s + (10.6 + 18.3i)4-s + 71.5i·8-s + (−57.8 + 33.3i)11-s + (−108. + 187. i)16-s − 361.·22-s + (108. + 62.5i)23-s + (62.5 + 108. i)25-s − 69.7i·29-s + (−519. + 300. i)32-s + (−5.29 + 9.16i)37-s + 534.·43-s + (−1.22e3 − 708. i)44-s + (338. + 585. i)46-s + 675. i·50-s + ⋯
L(s)  = 1  + (1.65 + 0.955i)2-s + (1.32 + 2.29i)4-s + 3.16i·8-s + (−1.58 + 0.915i)11-s + (−1.69 + 2.93i)16-s − 3.49·22-s + (0.982 + 0.567i)23-s + (0.5 + 0.866i)25-s − 0.446i·29-s + (−2.87 + 1.65i)32-s + (−0.0235 + 0.0407i)37-s + 1.89·43-s + (−4.20 − 2.42i)44-s + (1.08 + 1.87i)46-s + 1.91i·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 - 0.360i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.932 - 0.360i$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ -0.932 - 0.360i)\)

Particular Values

\(L(2)\) \(\approx\) \(4.360850055\)
\(L(\frac12)\) \(\approx\) \(4.360850055\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-4.68 - 2.70i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (57.8 - 33.3i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 2.19e3T^{2} \)
17 \( 1 + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-108. - 62.5i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 69.7iT - 2.43e4T^{2} \)
31 \( 1 + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (5.29 - 9.16i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 6.89e4T^{2} \)
43 \( 1 - 534.T + 7.95e4T^{2} \)
47 \( 1 + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (56.7 - 32.7i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (370 + 640. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 205. iT - 3.57e5T^{2} \)
73 \( 1 + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-692 + 1.19e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 5.71e5T^{2} \)
89 \( 1 + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 9.12e5T^{2} \)
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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

−11.38915463113874741979526221150, −10.50173772789637869778895853980, −9.044343236639076363045784420027, −7.66997957621608026562853918490, −7.41988748850852321715065770496, −6.21656280112794110742874543227, −5.23369567322174173000987870698, −4.65972367719101769526996741256, −3.36487408139336089601904711426, −2.35128838463964984595785767169, 0.77523765592600669712466630788, 2.44451244125646913323047823505, 3.12947029377546562278987199895, 4.40235137027345141033191651402, 5.28531184918336309147345275997, 6.03423705039728149898894969415, 7.18393800543960497318275539856, 8.557386966234249718249222617937, 9.932798285879136351714334259520, 10.81748469137185840943013059838

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