Properties

Label 2-7728-1.1-c1-0-120
Degree $2$
Conductor $7728$
Sign $-1$
Analytic cond. $61.7083$
Root an. cond. $7.85546$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2.93·5-s + 7-s + 9-s + 3.68·11-s − 6.61·13-s − 2.93·15-s − 4.18·17-s − 1.17·19-s − 21-s + 23-s + 3.61·25-s − 27-s + 1.68·29-s + 1.17·31-s − 3.68·33-s + 2.93·35-s − 2.31·37-s + 6.61·39-s − 7.17·41-s + 2.74·43-s + 2.93·45-s − 12.2·47-s + 49-s + 4.18·51-s − 13.1·53-s + 10.8·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.31·5-s + 0.377·7-s + 0.333·9-s + 1.10·11-s − 1.83·13-s − 0.757·15-s − 1.01·17-s − 0.269·19-s − 0.218·21-s + 0.208·23-s + 0.723·25-s − 0.192·27-s + 0.312·29-s + 0.210·31-s − 0.640·33-s + 0.496·35-s − 0.381·37-s + 1.05·39-s − 1.12·41-s + 0.418·43-s + 0.437·45-s − 1.78·47-s + 0.142·49-s + 0.586·51-s − 1.81·53-s + 1.45·55-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: no
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 \( 1 - 2.93T + 5T^{2} \)
11 \( 1 - 3.68T + 11T^{2} \)
13 \( 1 + 6.61T + 13T^{2} \)
17 \( 1 + 4.18T + 17T^{2} \)
19 \( 1 + 1.17T + 19T^{2} \)
29 \( 1 - 1.68T + 29T^{2} \)
31 \( 1 - 1.17T + 31T^{2} \)
37 \( 1 + 2.31T + 37T^{2} \)
41 \( 1 + 7.17T + 41T^{2} \)
43 \( 1 - 2.74T + 43T^{2} \)
47 \( 1 + 12.2T + 47T^{2} \)
53 \( 1 + 13.1T + 53T^{2} \)
59 \( 1 - 3.06T + 59T^{2} \)
61 \( 1 + 0.745T + 61T^{2} \)
67 \( 1 + 8.46T + 67T^{2} \)
71 \( 1 - 11.5T + 71T^{2} \)
73 \( 1 + 10.3T + 73T^{2} \)
79 \( 1 - 6.18T + 79T^{2} \)
83 \( 1 + 1.33T + 83T^{2} \)
89 \( 1 - 6.14T + 89T^{2} \)
97 \( 1 + 6.69T + 97T^{2} \)
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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

−7.31904438803778583503046812514, −6.55359542014422620599201444227, −6.36353945236985926590843048538, −5.27137477934515055466974454360, −4.90296606312278505527734519319, −4.17031416480320096000324276464, −2.92727112193938768399828031311, −2.04881613349570059573026247163, −1.46188195437792208920578892405, 0, 1.46188195437792208920578892405, 2.04881613349570059573026247163, 2.92727112193938768399828031311, 4.17031416480320096000324276464, 4.90296606312278505527734519319, 5.27137477934515055466974454360, 6.36353945236985926590843048538, 6.55359542014422620599201444227, 7.31904438803778583503046812514

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