Properties

Label 2-7800-1.1-c1-0-77
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 + 1.65·7-s + 9-s − 2.94·11-s + 13-s − 1.46·17-s − 1.65·21-s − 0.532·23-s − 27-s + 5.70·29-s + 2.94·33-s − 8.77·37-s − 39-s + 1.23·41-s + 1.70·43-s − 2.70·47-s − 4.25·49-s + 1.46·51-s + 8.77·53-s − 3.83·59-s − 0.241·61-s + 1.65·63-s − 2.58·67-s + 0.532·69-s − 2.55·71-s − 0.188·73-s − 4.88·77-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.625·7-s + 0.333·9-s − 0.888·11-s + 0.277·13-s − 0.355·17-s − 0.361·21-s − 0.111·23-s − 0.192·27-s + 1.06·29-s + 0.513·33-s − 1.44·37-s − 0.160·39-s + 0.193·41-s + 0.260·43-s − 0.394·47-s − 0.608·49-s + 0.205·51-s + 1.20·53-s − 0.498·59-s − 0.0308·61-s + 0.208·63-s − 0.315·67-s + 0.0641·69-s − 0.302·71-s − 0.0220·73-s − 0.556·77-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 \( 1 - 1.65T + 7T^{2} \)
11 \( 1 + 2.94T + 11T^{2} \)
17 \( 1 + 1.46T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 0.532T + 23T^{2} \)
29 \( 1 - 5.70T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 8.77T + 37T^{2} \)
41 \( 1 - 1.23T + 41T^{2} \)
43 \( 1 - 1.70T + 43T^{2} \)
47 \( 1 + 2.70T + 47T^{2} \)
53 \( 1 - 8.77T + 53T^{2} \)
59 \( 1 + 3.83T + 59T^{2} \)
61 \( 1 + 0.241T + 61T^{2} \)
67 \( 1 + 2.58T + 67T^{2} \)
71 \( 1 + 2.55T + 71T^{2} \)
73 \( 1 + 0.188T + 73T^{2} \)
79 \( 1 + 11.0T + 79T^{2} \)
83 \( 1 - 7.91T + 83T^{2} \)
89 \( 1 - 15.9T + 89T^{2} \)
97 \( 1 + 16.8T + 97T^{2} \)
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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

−7.47828775501774590323769849178, −6.81353155862705314190777315565, −6.08267057850894735255824026357, −5.33051518364156235967297098198, −4.82320837508619973334379590608, −4.07509907544541204728295473345, −3.08130469783594818068994250219, −2.16575678266499847351456010122, −1.22000292364139350708539200334, 0, 1.22000292364139350708539200334, 2.16575678266499847351456010122, 3.08130469783594818068994250219, 4.07509907544541204728295473345, 4.82320837508619973334379590608, 5.33051518364156235967297098198, 6.08267057850894735255824026357, 6.81353155862705314190777315565, 7.47828775501774590323769849178

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