Properties

Label 2-7800-1.1-c1-0-2
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2.82·7-s + 9-s − 6.46·11-s + 13-s − 3.49·17-s − 2.21·19-s + 2.82·21-s + 7.14·23-s − 27-s + 4.82·29-s − 9.65·31-s + 6.46·33-s − 6.93·37-s − 39-s − 0.148·41-s − 3.03·43-s − 6.79·47-s + 1.00·49-s + 3.49·51-s − 1.70·53-s + 2.21·57-s − 6.46·59-s + 2.66·61-s − 2.82·63-s + 7.70·67-s − 7.14·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.06·7-s + 0.333·9-s − 1.95·11-s + 0.277·13-s − 0.847·17-s − 0.507·19-s + 0.617·21-s + 1.49·23-s − 0.192·27-s + 0.896·29-s − 1.73·31-s + 1.12·33-s − 1.14·37-s − 0.160·39-s − 0.0232·41-s − 0.463·43-s − 0.990·47-s + 0.142·49-s + 0.489·51-s − 0.233·53-s + 0.292·57-s − 0.842·59-s + 0.341·61-s − 0.356·63-s + 0.941·67-s − 0.860·69-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: \(0\)
Selberg data: \((2,\ 7800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4251972290\)
\(L(\frac12)\) \(\approx\) \(0.4251972290\)
\(L(\frac{3}{2})\) 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 \( 1 + 6.46T + 11T^{2} \)
17 \( 1 + 3.49T + 17T^{2} \)
19 \( 1 + 2.21T + 19T^{2} \)
23 \( 1 - 7.14T + 23T^{2} \)
29 \( 1 - 4.82T + 29T^{2} \)
31 \( 1 + 9.65T + 31T^{2} \)
37 \( 1 + 6.93T + 37T^{2} \)
41 \( 1 + 0.148T + 41T^{2} \)
43 \( 1 + 3.03T + 43T^{2} \)
47 \( 1 + 6.79T + 47T^{2} \)
53 \( 1 + 1.70T + 53T^{2} \)
59 \( 1 + 6.46T + 59T^{2} \)
61 \( 1 - 2.66T + 61T^{2} \)
67 \( 1 - 7.70T + 67T^{2} \)
71 \( 1 + 11.0T + 71T^{2} \)
73 \( 1 + 5.76T + 73T^{2} \)
79 \( 1 + 10.0T + 79T^{2} \)
83 \( 1 - 2.27T + 83T^{2} \)
89 \( 1 + 9.21T + 89T^{2} \)
97 \( 1 - 8.11T + 97T^{2} \)
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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

−7.74421845891849322469837905886, −7.00633825628920895855688919578, −6.56326445858322964847232113356, −5.70289211048620456028851775400, −5.14133920779470504053840225042, −4.47810269452771585981505890072, −3.35298968027386971958932648160, −2.82235567713555755904745936845, −1.79002846068794539973371902259, −0.31707819442204257499110134278, 0.31707819442204257499110134278, 1.79002846068794539973371902259, 2.82235567713555755904745936845, 3.35298968027386971958932648160, 4.47810269452771585981505890072, 5.14133920779470504053840225042, 5.70289211048620456028851775400, 6.56326445858322964847232113356, 7.00633825628920895855688919578, 7.74421845891849322469837905886

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