Properties

Label 2-7800-1.1-c1-0-113
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

−7.62298124695175213327211594769, −7.01473243499615853878854932897, −6.10339624602389522349299584395, −5.27065723274372976964573113837, −4.60141199254077946656637275057, −4.08931592146035194875149623655, −2.99656402603008311346640098488, −2.08616424119685487402809240560, −1.62830310060678144196861963437, 0, 1.62830310060678144196861963437, 2.08616424119685487402809240560, 2.99656402603008311346640098488, 4.08931592146035194875149623655, 4.60141199254077946656637275057, 5.27065723274372976964573113837, 6.10339624602389522349299584395, 7.01473243499615853878854932897, 7.62298124695175213327211594769

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