Properties

Label 2-4928-1.1-c1-0-44
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  + 4·5-s − 7-s − 3·9-s + 11-s − 2·13-s − 4·17-s + 6·19-s + 4·23-s + 11·25-s + 2·29-s − 2·31-s − 4·35-s − 10·37-s + 4·41-s + 8·43-s − 12·45-s + 2·47-s + 49-s − 6·53-s + 4·55-s + 12·59-s + 14·61-s + 3·63-s − 8·65-s + 12·67-s − 8·71-s + 4·73-s + ⋯
L(s)  = 1  + 1.78·5-s − 0.377·7-s − 9-s + 0.301·11-s − 0.554·13-s − 0.970·17-s + 1.37·19-s + 0.834·23-s + 11/5·25-s + 0.371·29-s − 0.359·31-s − 0.676·35-s − 1.64·37-s + 0.624·41-s + 1.21·43-s − 1.78·45-s + 0.291·47-s + 1/7·49-s − 0.824·53-s + 0.539·55-s + 1.56·59-s + 1.79·61-s + 0.377·63-s − 0.992·65-s + 1.46·67-s − 0.949·71-s + 0.468·73-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: $\chi_{4928} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4928,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.504821876\)
\(L(\frac12)\) \(\approx\) \(2.504821876\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
5 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−8.596108761021651344869700453421, −7.29999456543540483713986491118, −6.75534837172482153142827818885, −6.00214639283948454302408706475, −5.40821631418111626955625209230, −4.89204912487217343955499873034, −3.53751482899012863138118644990, −2.66919692702130451505800501099, −2.09312337546441837636968536211, −0.868869516443346125712398445012, 0.868869516443346125712398445012, 2.09312337546441837636968536211, 2.66919692702130451505800501099, 3.53751482899012863138118644990, 4.89204912487217343955499873034, 5.40821631418111626955625209230, 6.00214639283948454302408706475, 6.75534837172482153142827818885, 7.29999456543540483713986491118, 8.596108761021651344869700453421

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