Properties

Label 2-4928-1.1-c1-0-39
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s + 7-s − 2·9-s − 11-s + 15-s − 2·17-s + 2·19-s + 21-s − 7·23-s − 4·25-s − 5·27-s + 10·29-s + 7·31-s − 33-s + 35-s + 9·37-s − 2·41-s + 4·43-s − 2·45-s + 8·47-s + 49-s − 2·51-s − 2·53-s − 55-s + 2·57-s + 15·59-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 0.377·7-s − 2/3·9-s − 0.301·11-s + 0.258·15-s − 0.485·17-s + 0.458·19-s + 0.218·21-s − 1.45·23-s − 4/5·25-s − 0.962·27-s + 1.85·29-s + 1.25·31-s − 0.174·33-s + 0.169·35-s + 1.47·37-s − 0.312·41-s + 0.609·43-s − 0.298·45-s + 1.16·47-s + 1/7·49-s − 0.280·51-s − 0.274·53-s − 0.134·55-s + 0.264·57-s + 1.95·59-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: \(0\)
Selberg data: \((2,\ 4928,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.477033017\)
\(L(\frac12)\) \(\approx\) \(2.477033017\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
5 \( 1 - T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 15 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 11 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−8.164400351857759128500212543992, −7.87404104579587702604894134834, −6.76510153951106529886427450825, −6.04956296563046921215939243259, −5.42136245461813480881489585872, −4.49170145379077500258866260165, −3.73442025194138113182506034259, −2.57891879893810287667817873696, −2.25395729266732590260667429868, −0.830336992343469057674078215670, 0.830336992343469057674078215670, 2.25395729266732590260667429868, 2.57891879893810287667817873696, 3.73442025194138113182506034259, 4.49170145379077500258866260165, 5.42136245461813480881489585872, 6.04956296563046921215939243259, 6.76510153951106529886427450825, 7.87404104579587702604894134834, 8.164400351857759128500212543992

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