Properties

Label 2-5415-1.1-c1-0-163
Degree $2$
Conductor $5415$
Sign $-1$
Analytic cond. $43.2389$
Root an. cond. $6.57563$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73·2-s − 3-s + 0.999·4-s + 5-s + 1.73·6-s + 0.732·7-s + 1.73·8-s + 9-s − 1.73·10-s + 1.26·11-s − 0.999·12-s + 2.73·13-s − 1.26·14-s − 15-s − 5·16-s − 1.73·18-s + 0.999·20-s − 0.732·21-s − 2.19·22-s − 3.46·23-s − 1.73·24-s + 25-s − 4.73·26-s − 27-s + 0.732·28-s + 2.19·29-s + 1.73·30-s + ⋯
L(s)  = 1  − 1.22·2-s − 0.577·3-s + 0.499·4-s + 0.447·5-s + 0.707·6-s + 0.276·7-s + 0.612·8-s + 0.333·9-s − 0.547·10-s + 0.382·11-s − 0.288·12-s + 0.757·13-s − 0.338·14-s − 0.258·15-s − 1.25·16-s − 0.408·18-s + 0.223·20-s − 0.159·21-s − 0.468·22-s − 0.722·23-s − 0.353·24-s + 0.200·25-s − 0.928·26-s − 0.192·27-s + 0.138·28-s + 0.407·29-s + 0.316·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5415\)    =    \(3 \cdot 5 \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(43.2389\)
Root analytic conductor: \(6.57563\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5415,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 - T \)
19 \( 1 \)
good2 \( 1 + 1.73T + 2T^{2} \)
7 \( 1 - 0.732T + 7T^{2} \)
11 \( 1 - 1.26T + 11T^{2} \)
13 \( 1 - 2.73T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
23 \( 1 + 3.46T + 23T^{2} \)
29 \( 1 - 2.19T + 29T^{2} \)
31 \( 1 - 4.92T + 31T^{2} \)
37 \( 1 + 4.19T + 37T^{2} \)
41 \( 1 + 4.73T + 41T^{2} \)
43 \( 1 + 6.19T + 43T^{2} \)
47 \( 1 - 3.46T + 47T^{2} \)
53 \( 1 - 2.53T + 53T^{2} \)
59 \( 1 + 9.46T + 59T^{2} \)
61 \( 1 + 13.4T + 61T^{2} \)
67 \( 1 + 8T + 67T^{2} \)
71 \( 1 + 16.3T + 71T^{2} \)
73 \( 1 + 3.07T + 73T^{2} \)
79 \( 1 + 2.92T + 79T^{2} \)
83 \( 1 - 0.928T + 83T^{2} \)
89 \( 1 + 7.26T + 89T^{2} \)
97 \( 1 + 4.19T + 97T^{2} \)
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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

−7.961071333282791354513617030432, −7.22032402910208055767803203691, −6.44528184226955045317251295393, −5.90051746416864977108158047715, −4.86827880212800519922536687528, −4.28718030625487614798127875633, −3.12943770216453842900956451971, −1.79933735172390308636248659644, −1.26012420311168812593244740108, 0, 1.26012420311168812593244740108, 1.79933735172390308636248659644, 3.12943770216453842900956451971, 4.28718030625487614798127875633, 4.86827880212800519922536687528, 5.90051746416864977108158047715, 6.44528184226955045317251295393, 7.22032402910208055767803203691, 7.961071333282791354513617030432

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