Properties

Label 2-7728-1.1-c1-0-93
Degree $2$
Conductor $7728$
Sign $-1$
Analytic cond. $61.7083$
Root an. cond. $7.85546$
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 + 7-s + 9-s + 2·11-s − 6·13-s + 2·17-s + 6·19-s − 21-s − 23-s − 5·25-s − 27-s − 6·29-s − 2·33-s + 6·39-s + 6·41-s + 6·43-s − 8·47-s + 49-s − 2·51-s − 4·53-s − 6·57-s − 8·61-s + 63-s + 2·67-s + 69-s − 2·73-s + 5·75-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.603·11-s − 1.66·13-s + 0.485·17-s + 1.37·19-s − 0.218·21-s − 0.208·23-s − 25-s − 0.192·27-s − 1.11·29-s − 0.348·33-s + 0.960·39-s + 0.937·41-s + 0.914·43-s − 1.16·47-s + 1/7·49-s − 0.280·51-s − 0.549·53-s − 0.794·57-s − 1.02·61-s + 0.125·63-s + 0.244·67-s + 0.120·69-s − 0.234·73-s + 0.577·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7728\)    =    \(2^{4} \cdot 3 \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(61.7083\)
Root analytic conductor: \(7.85546\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{7728} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7728,\ (\ :1/2),\ -1)\)

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

−7.60839998497940622646495106389, −6.92319938252210548801341388292, −5.98308017575993809356305078263, −5.44418806943124157061745404643, −4.77641761339239800879487035357, −4.04846806507089455955874915378, −3.13926693599211134068390463128, −2.13714269448030660340735657450, −1.22951817597761996846268232936, 0, 1.22951817597761996846268232936, 2.13714269448030660340735657450, 3.13926693599211134068390463128, 4.04846806507089455955874915378, 4.77641761339239800879487035357, 5.44418806943124157061745404643, 5.98308017575993809356305078263, 6.92319938252210548801341388292, 7.60839998497940622646495106389

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