Properties

Label 2-4928-1.1-c1-0-62
Degree $2$
Conductor $4928$
Sign $-1$
Analytic cond. $39.3502$
Root an. cond. $6.27298$
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  − 2·3-s − 2·5-s + 7-s + 9-s + 11-s + 4·15-s + 4·17-s − 4·19-s − 2·21-s − 4·23-s − 25-s + 4·27-s − 2·29-s − 2·31-s − 2·33-s − 2·35-s + 6·37-s + 4·41-s + 4·43-s − 2·45-s + 2·47-s + 49-s − 8·51-s − 2·53-s − 2·55-s + 8·57-s + 6·59-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s + 0.301·11-s + 1.03·15-s + 0.970·17-s − 0.917·19-s − 0.436·21-s − 0.834·23-s − 1/5·25-s + 0.769·27-s − 0.371·29-s − 0.359·31-s − 0.348·33-s − 0.338·35-s + 0.986·37-s + 0.624·41-s + 0.609·43-s − 0.298·45-s + 0.291·47-s + 1/7·49-s − 1.12·51-s − 0.274·53-s − 0.269·55-s + 1.05·57-s + 0.781·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4928\)    =    \(2^{6} \cdot 7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(39.3502\)
Root analytic conductor: \(6.27298\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4928,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 - T \)
11 \( 1 - T \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−7.81471559919429316687239101837, −7.24015489942259604405440552497, −6.22474682847289765271955059501, −5.84573150094526730078225656213, −4.95890967095362737510025794037, −4.23076734373564530341685680553, −3.57643311630839300149423314898, −2.31814487044434135855216314443, −1.05262565113522034371033309180, 0, 1.05262565113522034371033309180, 2.31814487044434135855216314443, 3.57643311630839300149423314898, 4.23076734373564530341685680553, 4.95890967095362737510025794037, 5.84573150094526730078225656213, 6.22474682847289765271955059501, 7.24015489942259604405440552497, 7.81471559919429316687239101837

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