Properties

Label 2-7800-1.1-c1-0-104
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 + 2.82·7-s + 9-s − 6.46·11-s − 13-s + 3.49·17-s − 2.21·19-s + 2.82·21-s − 7.14·23-s + 27-s + 4.82·29-s − 9.65·31-s − 6.46·33-s + 6.93·37-s − 39-s − 0.148·41-s + 3.03·43-s + 6.79·47-s + 1.00·49-s + 3.49·51-s + 1.70·53-s − 2.21·57-s − 6.46·59-s + 2.66·61-s + 2.82·63-s − 7.70·67-s − 7.14·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.06·7-s + 0.333·9-s − 1.95·11-s − 0.277·13-s + 0.847·17-s − 0.507·19-s + 0.617·21-s − 1.49·23-s + 0.192·27-s + 0.896·29-s − 1.73·31-s − 1.12·33-s + 1.14·37-s − 0.160·39-s − 0.0232·41-s + 0.463·43-s + 0.990·47-s + 0.142·49-s + 0.489·51-s + 0.233·53-s − 0.292·57-s − 0.842·59-s + 0.341·61-s + 0.356·63-s − 0.941·67-s − 0.860·69-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 \( 1 - 2.82T + 7T^{2} \)
11 \( 1 + 6.46T + 11T^{2} \)
17 \( 1 - 3.49T + 17T^{2} \)
19 \( 1 + 2.21T + 19T^{2} \)
23 \( 1 + 7.14T + 23T^{2} \)
29 \( 1 - 4.82T + 29T^{2} \)
31 \( 1 + 9.65T + 31T^{2} \)
37 \( 1 - 6.93T + 37T^{2} \)
41 \( 1 + 0.148T + 41T^{2} \)
43 \( 1 - 3.03T + 43T^{2} \)
47 \( 1 - 6.79T + 47T^{2} \)
53 \( 1 - 1.70T + 53T^{2} \)
59 \( 1 + 6.46T + 59T^{2} \)
61 \( 1 - 2.66T + 61T^{2} \)
67 \( 1 + 7.70T + 67T^{2} \)
71 \( 1 + 11.0T + 71T^{2} \)
73 \( 1 - 5.76T + 73T^{2} \)
79 \( 1 + 10.0T + 79T^{2} \)
83 \( 1 + 2.27T + 83T^{2} \)
89 \( 1 + 9.21T + 89T^{2} \)
97 \( 1 + 8.11T + 97T^{2} \)
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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

−7.72790062672760608738542524216, −7.13671775914136178670613300991, −5.87493998444514504985168377333, −5.45848866272016351322494616666, −4.62869307386204083014390507174, −4.01451061242531986994747150452, −2.89243853343559977716115035950, −2.34294414123532789363412968147, −1.46401820076699683352339139584, 0, 1.46401820076699683352339139584, 2.34294414123532789363412968147, 2.89243853343559977716115035950, 4.01451061242531986994747150452, 4.62869307386204083014390507174, 5.45848866272016351322494616666, 5.87493998444514504985168377333, 7.13671775914136178670613300991, 7.72790062672760608738542524216

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