Properties

Label 2-7800-1.1-c1-0-82
Degree $2$
Conductor $7800$
Sign $-1$
Analytic cond. $62.2833$
Root an. cond. $7.89197$
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  + 3-s − 2.82·7-s + 9-s − 2.97·11-s − 13-s + 1.44·17-s + 4.17·19-s − 2.82·21-s + 6.21·23-s + 27-s − 0.828·29-s + 1.65·31-s − 2.97·33-s − 0.0418·37-s − 39-s − 4.36·41-s − 8.99·43-s − 4.40·47-s + 1.00·49-s + 1.44·51-s − 6.72·53-s + 4.17·57-s − 2.97·59-s + 6.27·61-s − 2.82·63-s + 0.727·67-s + 6.21·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.06·7-s + 0.333·9-s − 0.898·11-s − 0.277·13-s + 0.350·17-s + 0.956·19-s − 0.617·21-s + 1.29·23-s + 0.192·27-s − 0.153·29-s + 0.297·31-s − 0.518·33-s − 0.00688·37-s − 0.160·39-s − 0.681·41-s − 1.37·43-s − 0.642·47-s + 0.142·49-s + 0.202·51-s − 0.924·53-s + 0.552·57-s − 0.387·59-s + 0.803·61-s − 0.356·63-s + 0.0888·67-s + 0.747·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7800\)    =    \(2^{3} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(62.2833\)
Root analytic conductor: \(7.89197\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7800,\ (\ :1/2),\ -1)\)

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 \( 1 \)
13 \( 1 + T \)
good7 \( 1 + 2.82T + 7T^{2} \)
11 \( 1 + 2.97T + 11T^{2} \)
17 \( 1 - 1.44T + 17T^{2} \)
19 \( 1 - 4.17T + 19T^{2} \)
23 \( 1 - 6.21T + 23T^{2} \)
29 \( 1 + 0.828T + 29T^{2} \)
31 \( 1 - 1.65T + 31T^{2} \)
37 \( 1 + 0.0418T + 37T^{2} \)
41 \( 1 + 4.36T + 41T^{2} \)
43 \( 1 + 8.99T + 43T^{2} \)
47 \( 1 + 4.40T + 47T^{2} \)
53 \( 1 + 6.72T + 53T^{2} \)
59 \( 1 + 2.97T + 59T^{2} \)
61 \( 1 - 6.27T + 61T^{2} \)
67 \( 1 - 0.727T + 67T^{2} \)
71 \( 1 + 8.32T + 71T^{2} \)
73 \( 1 + 6.87T + 73T^{2} \)
79 \( 1 - 6.11T + 79T^{2} \)
83 \( 1 - 14.7T + 83T^{2} \)
89 \( 1 - 14.7T + 89T^{2} \)
97 \( 1 + 6.78T + 97T^{2} \)
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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

−7.58495785378507056843354832705, −6.85180442945633627243905087711, −6.28027325806925293123811016742, −5.23796987539083065825447717412, −4.85163003984853616928655857133, −3.57461700295822538598409860490, −3.16689340016298290364426851294, −2.47301913613901996225835688300, −1.28687868872561569782711769978, 0, 1.28687868872561569782711769978, 2.47301913613901996225835688300, 3.16689340016298290364426851294, 3.57461700295822538598409860490, 4.85163003984853616928655857133, 5.23796987539083065825447717412, 6.28027325806925293123811016742, 6.85180442945633627243905087711, 7.58495785378507056843354832705

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