Properties

Label 2-7800-1.1-c1-0-92
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 − 0.476·7-s + 9-s − 3.34·11-s − 13-s − 17-s − 6.77·19-s − 0.476·21-s + 4.77·23-s + 27-s + 7.59·29-s + 7.93·31-s − 3.34·33-s − 7.82·37-s − 39-s + 9.46·41-s − 1.82·43-s + 9.06·47-s − 6.77·49-s − 51-s − 9.50·53-s − 6.77·57-s + 3.16·59-s − 9.59·61-s − 0.476·63-s + 1.34·67-s + 4.77·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.180·7-s + 0.333·9-s − 1.00·11-s − 0.277·13-s − 0.242·17-s − 1.55·19-s − 0.103·21-s + 0.995·23-s + 0.192·27-s + 1.41·29-s + 1.42·31-s − 0.582·33-s − 1.28·37-s − 0.160·39-s + 1.47·41-s − 0.277·43-s + 1.32·47-s − 0.967·49-s − 0.140·51-s − 1.30·53-s − 0.897·57-s + 0.411·59-s − 1.22·61-s − 0.0600·63-s + 0.164·67-s + 0.574·69-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 \( 1 + 0.476T + 7T^{2} \)
11 \( 1 + 3.34T + 11T^{2} \)
17 \( 1 + T + 17T^{2} \)
19 \( 1 + 6.77T + 19T^{2} \)
23 \( 1 - 4.77T + 23T^{2} \)
29 \( 1 - 7.59T + 29T^{2} \)
31 \( 1 - 7.93T + 31T^{2} \)
37 \( 1 + 7.82T + 37T^{2} \)
41 \( 1 - 9.46T + 41T^{2} \)
43 \( 1 + 1.82T + 43T^{2} \)
47 \( 1 - 9.06T + 47T^{2} \)
53 \( 1 + 9.50T + 53T^{2} \)
59 \( 1 - 3.16T + 59T^{2} \)
61 \( 1 + 9.59T + 61T^{2} \)
67 \( 1 - 1.34T + 67T^{2} \)
71 \( 1 + 4.86T + 71T^{2} \)
73 \( 1 - 14.4T + 73T^{2} \)
79 \( 1 + 2.17T + 79T^{2} \)
83 \( 1 + 9.16T + 83T^{2} \)
89 \( 1 + 10.5T + 89T^{2} \)
97 \( 1 + 7.04T + 97T^{2} \)
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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

−7.64836977191603915429201226211, −6.74826729799244326616851944656, −6.34684616556637364500724730651, −5.27231316095327368916784673723, −4.64394502821784250997703741371, −3.96455204586088306593940314204, −2.75294442567611409536773755567, −2.61398239626680768403055755465, −1.33759323868721965817113338273, 0, 1.33759323868721965817113338273, 2.61398239626680768403055755465, 2.75294442567611409536773755567, 3.96455204586088306593940314204, 4.64394502821784250997703741371, 5.27231316095327368916784673723, 6.34684616556637364500724730651, 6.74826729799244326616851944656, 7.64836977191603915429201226211

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