Properties

Label 2-7728-1.1-c1-0-128
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 + 1.61·5-s + 7-s + 9-s + 2.23·11-s − 4.61·13-s + 1.61·15-s − 6.70·17-s − 5.47·19-s + 21-s + 23-s − 2.38·25-s + 27-s − 3.76·29-s + 6.70·31-s + 2.23·33-s + 1.61·35-s − 11·37-s − 4.61·39-s + 7.47·41-s − 0.618·43-s + 1.61·45-s + 2.76·47-s + 49-s − 6.70·51-s − 1.90·53-s + 3.61·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.723·5-s + 0.377·7-s + 0.333·9-s + 0.674·11-s − 1.28·13-s + 0.417·15-s − 1.62·17-s − 1.25·19-s + 0.218·21-s + 0.208·23-s − 0.476·25-s + 0.192·27-s − 0.698·29-s + 1.20·31-s + 0.389·33-s + 0.273·35-s − 1.80·37-s − 0.739·39-s + 1.16·41-s − 0.0942·43-s + 0.241·45-s + 0.403·47-s + 0.142·49-s − 0.939·51-s − 0.262·53-s + 0.487·55-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 \( 1 - 1.61T + 5T^{2} \)
11 \( 1 - 2.23T + 11T^{2} \)
13 \( 1 + 4.61T + 13T^{2} \)
17 \( 1 + 6.70T + 17T^{2} \)
19 \( 1 + 5.47T + 19T^{2} \)
29 \( 1 + 3.76T + 29T^{2} \)
31 \( 1 - 6.70T + 31T^{2} \)
37 \( 1 + 11T + 37T^{2} \)
41 \( 1 - 7.47T + 41T^{2} \)
43 \( 1 + 0.618T + 43T^{2} \)
47 \( 1 - 2.76T + 47T^{2} \)
53 \( 1 + 1.90T + 53T^{2} \)
59 \( 1 + 11.6T + 59T^{2} \)
61 \( 1 - 1.85T + 61T^{2} \)
67 \( 1 + 6.09T + 67T^{2} \)
71 \( 1 + 6.61T + 71T^{2} \)
73 \( 1 - 0.708T + 73T^{2} \)
79 \( 1 - 0.527T + 79T^{2} \)
83 \( 1 + 13.1T + 83T^{2} \)
89 \( 1 - 9.38T + 89T^{2} \)
97 \( 1 + 16.4T + 97T^{2} \)
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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

−7.49174185690278150122121328506, −6.81108386078542963819171176447, −6.27717675271568572783353313241, −5.38709796497648731496636267061, −4.49096009984607168165422525718, −4.14736443020660706056601097468, −2.88978472574462379918799319639, −2.18739715156204261957333036861, −1.62282548934540466802950009127, 0, 1.62282548934540466802950009127, 2.18739715156204261957333036861, 2.88978472574462379918799319639, 4.14736443020660706056601097468, 4.49096009984607168165422525718, 5.38709796497648731496636267061, 6.27717675271568572783353313241, 6.81108386078542963819171176447, 7.49174185690278150122121328506

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