Properties

Label 2-4620-1.1-c1-0-27
Degree $2$
Conductor $4620$
Sign $-1$
Analytic cond. $36.8908$
Root an. cond. $6.07378$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s + 7-s + 9-s − 11-s + 2·13-s + 15-s − 7·17-s + 3·19-s − 21-s + 3·23-s + 25-s − 27-s − 5·29-s + 33-s − 35-s + 2·37-s − 2·39-s + 43-s − 45-s + 8·47-s + 49-s + 7·51-s − 9·53-s + 55-s − 3·57-s − 9·59-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 0.258·15-s − 1.69·17-s + 0.688·19-s − 0.218·21-s + 0.625·23-s + 1/5·25-s − 0.192·27-s − 0.928·29-s + 0.174·33-s − 0.169·35-s + 0.328·37-s − 0.320·39-s + 0.152·43-s − 0.149·45-s + 1.16·47-s + 1/7·49-s + 0.980·51-s − 1.23·53-s + 0.134·55-s − 0.397·57-s − 1.17·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4620\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(36.8908\)
Root analytic conductor: \(6.07378\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4620,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 + T \)
good13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 5 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−7.78225500677182705297332048355, −7.30336231191537439889256836716, −6.46883887331436998124488785674, −5.78529533181259764992373514999, −4.90610131514061825327369965470, −4.34764133494520801888026569575, −3.44425184165626550095176281670, −2.37117777658916015617857375561, −1.26393328193811222436942412170, 0, 1.26393328193811222436942412170, 2.37117777658916015617857375561, 3.44425184165626550095176281670, 4.34764133494520801888026569575, 4.90610131514061825327369965470, 5.78529533181259764992373514999, 6.46883887331436998124488785674, 7.30336231191537439889256836716, 7.78225500677182705297332048355

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