Properties

Label 2-7800-1.1-c1-0-56
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 − 4.11·7-s + 9-s − 3.21·11-s − 13-s + 6.05·17-s − 2.90·19-s + 4.11·21-s + 3.65·23-s − 27-s − 0.377·29-s − 1.40·31-s + 3.21·33-s + 10.0·37-s + 39-s + 1.33·41-s + 6.57·43-s − 8.11·47-s + 9.95·49-s − 6.05·51-s + 4.09·53-s + 2.90·57-s − 14.2·59-s − 2.57·61-s − 4.11·63-s + 7.02·67-s − 3.65·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.55·7-s + 0.333·9-s − 0.969·11-s − 0.277·13-s + 1.46·17-s − 0.666·19-s + 0.898·21-s + 0.762·23-s − 0.192·27-s − 0.0701·29-s − 0.252·31-s + 0.559·33-s + 1.65·37-s + 0.160·39-s + 0.208·41-s + 1.00·43-s − 1.18·47-s + 1.42·49-s − 0.847·51-s + 0.562·53-s + 0.384·57-s − 1.85·59-s − 0.329·61-s − 0.518·63-s + 0.857·67-s − 0.440·69-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 \( 1 + 4.11T + 7T^{2} \)
11 \( 1 + 3.21T + 11T^{2} \)
17 \( 1 - 6.05T + 17T^{2} \)
19 \( 1 + 2.90T + 19T^{2} \)
23 \( 1 - 3.65T + 23T^{2} \)
29 \( 1 + 0.377T + 29T^{2} \)
31 \( 1 + 1.40T + 31T^{2} \)
37 \( 1 - 10.0T + 37T^{2} \)
41 \( 1 - 1.33T + 41T^{2} \)
43 \( 1 - 6.57T + 43T^{2} \)
47 \( 1 + 8.11T + 47T^{2} \)
53 \( 1 - 4.09T + 53T^{2} \)
59 \( 1 + 14.2T + 59T^{2} \)
61 \( 1 + 2.57T + 61T^{2} \)
67 \( 1 - 7.02T + 67T^{2} \)
71 \( 1 - 7.19T + 71T^{2} \)
73 \( 1 - 7.46T + 73T^{2} \)
79 \( 1 - 7.13T + 79T^{2} \)
83 \( 1 - 10.9T + 83T^{2} \)
89 \( 1 + 18.5T + 89T^{2} \)
97 \( 1 + 12.2T + 97T^{2} \)
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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

−7.53187686372427572615055812449, −6.67446114524337539152497094972, −6.14357637020982794866056435722, −5.51136156742285234112041417663, −4.81565279272108814951564343679, −3.84233490312395565961706473238, −3.09696383709189731672943090669, −2.42578313508821444493941527897, −0.988677189165820810713725797577, 0, 0.988677189165820810713725797577, 2.42578313508821444493941527897, 3.09696383709189731672943090669, 3.84233490312395565961706473238, 4.81565279272108814951564343679, 5.51136156742285234112041417663, 6.14357637020982794866056435722, 6.67446114524337539152497094972, 7.53187686372427572615055812449

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