Properties

Label 2-7800-1.1-c1-0-31
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 − 1.48·7-s + 9-s + 5.77·11-s + 13-s − 3·17-s + 2.29·19-s + 1.48·21-s + 4.29·23-s − 27-s + 6.55·29-s + 5.77·31-s − 5.77·33-s − 3.25·37-s − 39-s + 5.25·41-s − 5.25·43-s + 5.03·47-s − 4.80·49-s + 3·51-s + 3.29·53-s − 2.29·57-s − 5.03·59-s − 4.55·61-s − 1.48·63-s − 4.22·67-s − 4.29·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.559·7-s + 0.333·9-s + 1.74·11-s + 0.277·13-s − 0.727·17-s + 0.526·19-s + 0.323·21-s + 0.895·23-s − 0.192·27-s + 1.21·29-s + 1.03·31-s − 1.00·33-s − 0.535·37-s − 0.160·39-s + 0.820·41-s − 0.801·43-s + 0.734·47-s − 0.686·49-s + 0.420·51-s + 0.452·53-s − 0.304·57-s − 0.655·59-s − 0.582·61-s − 0.186·63-s − 0.516·67-s − 0.517·69-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

\(L(1)\) \(\approx\) \(1.826125302\)
\(L(\frac12)\) \(\approx\) \(1.826125302\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.48T + 7T^{2} \)
11 \( 1 - 5.77T + 11T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 - 2.29T + 19T^{2} \)
23 \( 1 - 4.29T + 23T^{2} \)
29 \( 1 - 6.55T + 29T^{2} \)
31 \( 1 - 5.77T + 31T^{2} \)
37 \( 1 + 3.25T + 37T^{2} \)
41 \( 1 - 5.25T + 41T^{2} \)
43 \( 1 + 5.25T + 43T^{2} \)
47 \( 1 - 5.03T + 47T^{2} \)
53 \( 1 - 3.29T + 53T^{2} \)
59 \( 1 + 5.03T + 59T^{2} \)
61 \( 1 + 4.55T + 61T^{2} \)
67 \( 1 + 4.22T + 67T^{2} \)
71 \( 1 + 12.8T + 71T^{2} \)
73 \( 1 - 12.8T + 73T^{2} \)
79 \( 1 + 1.25T + 79T^{2} \)
83 \( 1 - 12.0T + 83T^{2} \)
89 \( 1 - 13.5T + 89T^{2} \)
97 \( 1 + 6.96T + 97T^{2} \)
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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

−7.76459393495368403486237135869, −6.84030214290695697592810738560, −6.54523130366209625806727886228, −5.98514808863659217335385284837, −4.97232584937965596627538021390, −4.35949279828071887379529756808, −3.58865592329480564298293670290, −2.78448621832531156487853621400, −1.53369341376602720504193493637, −0.74892412139391629667838280183, 0.74892412139391629667838280183, 1.53369341376602720504193493637, 2.78448621832531156487853621400, 3.58865592329480564298293670290, 4.35949279828071887379529756808, 4.97232584937965596627538021390, 5.98514808863659217335385284837, 6.54523130366209625806727886228, 6.84030214290695697592810738560, 7.76459393495368403486237135869

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