Properties

Label 2-7800-1.1-c1-0-41
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 − 2.40·7-s + 9-s + 5.38·11-s + 13-s + 7.17·17-s + 7.79·19-s + 2.40·21-s + 2.40·23-s − 27-s + 4.97·29-s − 6.76·31-s − 5.38·33-s − 7.38·37-s − 39-s + 3.38·41-s + 11.5·43-s + 10.7·47-s − 1.19·49-s − 7.17·51-s − 1.43·53-s − 7.79·57-s + 5.94·59-s − 1.43·61-s − 2.40·63-s − 7.58·67-s − 2.40·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.910·7-s + 0.333·9-s + 1.62·11-s + 0.277·13-s + 1.73·17-s + 1.78·19-s + 0.525·21-s + 0.502·23-s − 0.192·27-s + 0.923·29-s − 1.21·31-s − 0.936·33-s − 1.21·37-s − 0.160·39-s + 0.528·41-s + 1.76·43-s + 1.56·47-s − 0.171·49-s − 1.00·51-s − 0.197·53-s − 1.03·57-s + 0.774·59-s − 0.183·61-s − 0.303·63-s − 0.926·67-s − 0.289·69-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

\(L(1)\) \(\approx\) \(2.040699284\)
\(L(\frac12)\) \(\approx\) \(2.040699284\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.40T + 7T^{2} \)
11 \( 1 - 5.38T + 11T^{2} \)
17 \( 1 - 7.17T + 17T^{2} \)
19 \( 1 - 7.79T + 19T^{2} \)
23 \( 1 - 2.40T + 23T^{2} \)
29 \( 1 - 4.97T + 29T^{2} \)
31 \( 1 + 6.76T + 31T^{2} \)
37 \( 1 + 7.38T + 37T^{2} \)
41 \( 1 - 3.38T + 41T^{2} \)
43 \( 1 - 11.5T + 43T^{2} \)
47 \( 1 - 10.7T + 47T^{2} \)
53 \( 1 + 1.43T + 53T^{2} \)
59 \( 1 - 5.94T + 59T^{2} \)
61 \( 1 + 1.43T + 61T^{2} \)
67 \( 1 + 7.58T + 67T^{2} \)
71 \( 1 + 2.61T + 71T^{2} \)
73 \( 1 + 7.02T + 73T^{2} \)
79 \( 1 - 2.61T + 79T^{2} \)
83 \( 1 + 1.94T + 83T^{2} \)
89 \( 1 - 0.618T + 89T^{2} \)
97 \( 1 + 9.22T + 97T^{2} \)
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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

−7.49514968367446964983524613759, −7.23783616809591309231491582036, −6.40893911343800686920452359440, −5.76087691844769913100860837969, −5.30008785360804659154846257976, −4.16795881891213329660893751011, −3.54123521895968945283985375720, −2.92917790231481529802574610130, −1.40455772732931851586494136309, −0.843583357669365926087712123350, 0.843583357669365926087712123350, 1.40455772732931851586494136309, 2.92917790231481529802574610130, 3.54123521895968945283985375720, 4.16795881891213329660893751011, 5.30008785360804659154846257976, 5.76087691844769913100860837969, 6.40893911343800686920452359440, 7.23783616809591309231491582036, 7.49514968367446964983524613759

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