Properties

Label 2-21e2-1.1-c5-0-66
Degree $2$
Conductor $441$
Sign $-1$
Analytic cond. $70.7292$
Root an. cond. $8.41006$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·2-s − 7·4-s + 94·5-s + 195·8-s − 470·10-s − 52·11-s + 770·13-s − 751·16-s − 2.02e3·17-s − 1.73e3·19-s − 658·20-s + 260·22-s + 576·23-s + 5.71e3·25-s − 3.85e3·26-s − 5.51e3·29-s − 6.33e3·31-s − 2.48e3·32-s + 1.01e4·34-s − 7.33e3·37-s + 8.66e3·38-s + 1.83e4·40-s − 3.26e3·41-s + 5.42e3·43-s + 364·44-s − 2.88e3·46-s + 864·47-s + ⋯
L(s)  = 1  − 0.883·2-s − 0.218·4-s + 1.68·5-s + 1.07·8-s − 1.48·10-s − 0.129·11-s + 1.26·13-s − 0.733·16-s − 1.69·17-s − 1.10·19-s − 0.367·20-s + 0.114·22-s + 0.227·23-s + 1.82·25-s − 1.11·26-s − 1.21·29-s − 1.18·31-s − 0.428·32-s + 1.49·34-s − 0.881·37-s + 0.972·38-s + 1.81·40-s − 0.303·41-s + 0.447·43-s + 0.0283·44-s − 0.200·46-s + 0.0570·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(70.7292\)
Root analytic conductor: \(8.41006\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 441,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 + 5 T + p^{5} T^{2} \)
5 \( 1 - 94 T + p^{5} T^{2} \)
11 \( 1 + 52 T + p^{5} T^{2} \)
13 \( 1 - 770 T + p^{5} T^{2} \)
17 \( 1 + 2022 T + p^{5} T^{2} \)
19 \( 1 + 1732 T + p^{5} T^{2} \)
23 \( 1 - 576 T + p^{5} T^{2} \)
29 \( 1 + 5518 T + p^{5} T^{2} \)
31 \( 1 + 6336 T + p^{5} T^{2} \)
37 \( 1 + 7338 T + p^{5} T^{2} \)
41 \( 1 + 3262 T + p^{5} T^{2} \)
43 \( 1 - 5420 T + p^{5} T^{2} \)
47 \( 1 - 864 T + p^{5} T^{2} \)
53 \( 1 + 4182 T + p^{5} T^{2} \)
59 \( 1 + 11220 T + p^{5} T^{2} \)
61 \( 1 - 45602 T + p^{5} T^{2} \)
67 \( 1 - 1396 T + p^{5} T^{2} \)
71 \( 1 + 18720 T + p^{5} T^{2} \)
73 \( 1 + 46362 T + p^{5} T^{2} \)
79 \( 1 - 97424 T + p^{5} T^{2} \)
83 \( 1 + 81228 T + p^{5} T^{2} \)
89 \( 1 + 3182 T + p^{5} T^{2} \)
97 \( 1 + 4914 T + p^{5} T^{2} \)
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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

−9.686879902953372506515996983686, −8.959289522120979125219220056616, −8.530524527063562654135362451645, −7.05665041472596775420207211064, −6.17916793218106919672180562051, −5.21700641444844076965709051319, −3.99393072620464901544562362410, −2.19698957695456137349275900442, −1.47948069039689251462319951770, 0, 1.47948069039689251462319951770, 2.19698957695456137349275900442, 3.99393072620464901544562362410, 5.21700641444844076965709051319, 6.17916793218106919672180562051, 7.05665041472596775420207211064, 8.530524527063562654135362451645, 8.959289522120979125219220056616, 9.686879902953372506515996983686

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