Properties

Label 2-7728-1.1-c1-0-111
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 − 0.167·5-s − 7-s + 9-s + 1.80·11-s − 3.63·13-s − 0.167·15-s + 4.13·17-s − 5.80·19-s − 21-s + 23-s − 4.97·25-s + 27-s + 5·29-s − 0.195·31-s + 1.80·33-s + 0.167·35-s − 3.33·37-s − 3.63·39-s − 4.94·41-s − 1.30·43-s − 0.167·45-s + 3.13·47-s + 49-s + 4.13·51-s + 3.30·53-s − 0.302·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.0748·5-s − 0.377·7-s + 0.333·9-s + 0.544·11-s − 1.00·13-s − 0.0432·15-s + 1.00·17-s − 1.33·19-s − 0.218·21-s + 0.208·23-s − 0.994·25-s + 0.192·27-s + 0.928·29-s − 0.0351·31-s + 0.314·33-s + 0.0283·35-s − 0.548·37-s − 0.582·39-s − 0.772·41-s − 0.198·43-s − 0.0249·45-s + 0.457·47-s + 0.142·49-s + 0.579·51-s + 0.454·53-s − 0.0407·55-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 \( 1 + 0.167T + 5T^{2} \)
11 \( 1 - 1.80T + 11T^{2} \)
13 \( 1 + 3.63T + 13T^{2} \)
17 \( 1 - 4.13T + 17T^{2} \)
19 \( 1 + 5.80T + 19T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 + 0.195T + 31T^{2} \)
37 \( 1 + 3.33T + 37T^{2} \)
41 \( 1 + 4.94T + 41T^{2} \)
43 \( 1 + 1.30T + 43T^{2} \)
47 \( 1 - 3.13T + 47T^{2} \)
53 \( 1 - 3.30T + 53T^{2} \)
59 \( 1 - 4.91T + 59T^{2} \)
61 \( 1 - 8.77T + 61T^{2} \)
67 \( 1 + 13.2T + 67T^{2} \)
71 \( 1 + 8.77T + 71T^{2} \)
73 \( 1 + 12.4T + 73T^{2} \)
79 \( 1 + 1.52T + 79T^{2} \)
83 \( 1 + 14.7T + 83T^{2} \)
89 \( 1 - 8.77T + 89T^{2} \)
97 \( 1 + 14.2T + 97T^{2} \)
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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

−7.44707178851839721215389747583, −6.97506283726966712610440506470, −6.19377945286188496315444339444, −5.44128282280585246691598326272, −4.52050624535094238869098194623, −3.91360949497471905638325146021, −3.07902687060187516817010928700, −2.33770775849185224575453048889, −1.38532754450748156164289082179, 0, 1.38532754450748156164289082179, 2.33770775849185224575453048889, 3.07902687060187516817010928700, 3.91360949497471905638325146021, 4.52050624535094238869098194623, 5.44128282280585246691598326272, 6.19377945286188496315444339444, 6.97506283726966712610440506470, 7.44707178851839721215389747583

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