Properties

Label 2-21e2-1.1-c3-0-32
Degree $2$
Conductor $441$
Sign $-1$
Analytic cond. $26.0198$
Root an. cond. $5.10096$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·4-s + 70·13-s + 64·16-s − 56·19-s − 125·25-s − 308·31-s + 110·37-s − 520·43-s − 560·52-s − 182·61-s − 512·64-s − 880·67-s − 1.19e3·73-s + 448·76-s + 884·79-s + 1.33e3·97-s + 1.00e3·100-s − 1.82e3·103-s − 646·109-s + ⋯
L(s)  = 1  − 4-s + 1.49·13-s + 16-s − 0.676·19-s − 25-s − 1.78·31-s + 0.488·37-s − 1.84·43-s − 1.49·52-s − 0.382·61-s − 64-s − 1.60·67-s − 1.90·73-s + 0.676·76-s + 1.25·79-s + 1.39·97-s + 100-s − 1.74·103-s − 0.567·109-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{441} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + p^{3} T^{2} \)
5 \( 1 + p^{3} T^{2} \)
11 \( 1 + p^{3} T^{2} \)
13 \( 1 - 70 T + p^{3} T^{2} \)
17 \( 1 + p^{3} T^{2} \)
19 \( 1 + 56 T + p^{3} T^{2} \)
23 \( 1 + p^{3} T^{2} \)
29 \( 1 + p^{3} T^{2} \)
31 \( 1 + 308 T + p^{3} T^{2} \)
37 \( 1 - 110 T + p^{3} T^{2} \)
41 \( 1 + p^{3} T^{2} \)
43 \( 1 + 520 T + p^{3} T^{2} \)
47 \( 1 + p^{3} T^{2} \)
53 \( 1 + p^{3} T^{2} \)
59 \( 1 + p^{3} T^{2} \)
61 \( 1 + 182 T + p^{3} T^{2} \)
67 \( 1 + 880 T + p^{3} T^{2} \)
71 \( 1 + p^{3} T^{2} \)
73 \( 1 + 1190 T + p^{3} T^{2} \)
79 \( 1 - 884 T + p^{3} T^{2} \)
83 \( 1 + p^{3} T^{2} \)
89 \( 1 + p^{3} T^{2} \)
97 \( 1 - 1330 T + p^{3} T^{2} \)
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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

−10.22817985803238738613607494828, −9.211112888137510527811358331321, −8.569740652483581810885164389919, −7.68699538159947646003388525207, −6.31927327676018381220493936485, −5.45495897269880459318633641846, −4.24962713348849248471173972282, −3.43152696950900144665279677676, −1.55867456761572882180639767698, 0, 1.55867456761572882180639767698, 3.43152696950900144665279677676, 4.24962713348849248471173972282, 5.45495897269880459318633641846, 6.31927327676018381220493936485, 7.68699538159947646003388525207, 8.569740652483581810885164389919, 9.211112888137510527811358331321, 10.22817985803238738613607494828

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