Properties

Label 2-624-1.1-c1-0-9
Degree $2$
Conductor $624$
Sign $-1$
Analytic cond. $4.98266$
Root an. cond. $2.23218$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·7-s + 9-s + 13-s − 6·17-s − 2·19-s + 2·21-s − 5·25-s − 27-s − 6·29-s − 2·31-s + 2·37-s − 39-s − 12·41-s + 4·43-s − 3·49-s + 6·51-s + 6·53-s + 2·57-s − 12·59-s + 2·61-s − 2·63-s + 10·67-s − 12·71-s + 14·73-s + 5·75-s − 8·79-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.755·7-s + 1/3·9-s + 0.277·13-s − 1.45·17-s − 0.458·19-s + 0.436·21-s − 25-s − 0.192·27-s − 1.11·29-s − 0.359·31-s + 0.328·37-s − 0.160·39-s − 1.87·41-s + 0.609·43-s − 3/7·49-s + 0.840·51-s + 0.824·53-s + 0.264·57-s − 1.56·59-s + 0.256·61-s − 0.251·63-s + 1.22·67-s − 1.42·71-s + 1.63·73-s + 0.577·75-s − 0.900·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-1$
Analytic conductor: \(4.98266\)
Root analytic conductor: \(2.23218\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 624,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
13 \( 1 - T \)
good5 \( 1 + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−10.21634971619086081222499193318, −9.385896632264966978634773161619, −8.531128945035369322451694692338, −7.32625330559643707687229250500, −6.49558938607564697804430447032, −5.75513068077913273540546960045, −4.55277073254635650297368891844, −3.54130855144350797063436114136, −2.01665432147547119953995139615, 0, 2.01665432147547119953995139615, 3.54130855144350797063436114136, 4.55277073254635650297368891844, 5.75513068077913273540546960045, 6.49558938607564697804430447032, 7.32625330559643707687229250500, 8.531128945035369322451694692338, 9.385896632264966978634773161619, 10.21634971619086081222499193318

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