Properties

Label 2-4928-1.1-c1-0-32
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  + 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: \(0\)
Selberg data: \((2,\ 4928,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.247635574\)
\(L(\frac12)\) \(\approx\) \(2.247635574\)
\(L(\frac{3}{2})\) 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

−8.036991817391048098082022022747, −7.79176991605327917560108987491, −7.13469277269583351110067947880, −6.11939333308679473761121471279, −5.26631240023283908554477437321, −4.36235253180583064087154778916, −3.34630364038171409907826355088, −3.21848782855454117736763082704, −2.11705595583072934562648531412, −0.76910974709403191668055663643, 0.76910974709403191668055663643, 2.11705595583072934562648531412, 3.21848782855454117736763082704, 3.34630364038171409907826355088, 4.36235253180583064087154778916, 5.26631240023283908554477437321, 6.11939333308679473761121471279, 7.13469277269583351110067947880, 7.79176991605327917560108987491, 8.036991817391048098082022022747

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