Properties

Degree $2$
Conductor $3630$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 5-s + 6-s + 3.73·7-s + 8-s + 9-s + 10-s + 12-s + 3·13-s + 3.73·14-s + 15-s + 16-s + 0.267·17-s + 18-s − 4.46·19-s + 20-s + 3.73·21-s − 4.73·23-s + 24-s + 25-s + 3·26-s + 27-s + 3.73·28-s − 3·29-s + 30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 1.41·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.288·12-s + 0.832·13-s + 0.997·14-s + 0.258·15-s + 0.250·16-s + 0.0649·17-s + 0.235·18-s − 1.02·19-s + 0.223·20-s + 0.814·21-s − 0.986·23-s + 0.204·24-s + 0.200·25-s + 0.588·26-s + 0.192·27-s + 0.705·28-s − 0.557·29-s + 0.182·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3630\)    =    \(2 \cdot 3 \cdot 5 \cdot 11^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{3630} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3630,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.019941541\)
\(L(\frac12)\) \(\approx\) \(5.019941541\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 - T \)
11 \( 1 \)
good7 \( 1 - 3.73T + 7T^{2} \)
13 \( 1 - 3T + 13T^{2} \)
17 \( 1 - 0.267T + 17T^{2} \)
19 \( 1 + 4.46T + 19T^{2} \)
23 \( 1 + 4.73T + 23T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 - 8.19T + 31T^{2} \)
37 \( 1 + 1.73T + 37T^{2} \)
41 \( 1 - 4.73T + 41T^{2} \)
43 \( 1 - 6.73T + 43T^{2} \)
47 \( 1 - 8.92T + 47T^{2} \)
53 \( 1 + 5.26T + 53T^{2} \)
59 \( 1 + 4.19T + 59T^{2} \)
61 \( 1 + 2.19T + 61T^{2} \)
67 \( 1 + 11.4T + 67T^{2} \)
71 \( 1 + 9.92T + 71T^{2} \)
73 \( 1 - 6.19T + 73T^{2} \)
79 \( 1 - 0.535T + 79T^{2} \)
83 \( 1 + 11.3T + 83T^{2} \)
89 \( 1 + 16.3T + 89T^{2} \)
97 \( 1 + 18.7T + 97T^{2} \)
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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

−8.387457645867617326135688056787, −7.910554230765527998978303499899, −7.07747027785791675460445911131, −6.09156048186245265312117252397, −5.60901656647989014352336919055, −4.40148008582044716569074615489, −4.23729570019971350028836290756, −2.94969540625736663893497656088, −2.08239217263590813845521553259, −1.33019107023578876903192351829, 1.33019107023578876903192351829, 2.08239217263590813845521553259, 2.94969540625736663893497656088, 4.23729570019971350028836290756, 4.40148008582044716569074615489, 5.60901656647989014352336919055, 6.09156048186245265312117252397, 7.07747027785791675460445911131, 7.910554230765527998978303499899, 8.387457645867617326135688056787

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