Properties

Label 2-7728-1.1-c1-0-73
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 3.17·5-s + 7-s + 9-s − 5.06·11-s + 4.07·13-s + 3.17·15-s + 4.22·17-s + 5.06·19-s + 21-s − 23-s + 5.07·25-s + 27-s + 6.68·29-s + 2.22·31-s − 5.06·33-s + 3.17·35-s − 1.91·37-s + 4.07·39-s − 1.37·41-s + 3.39·43-s + 3.17·45-s + 5.81·47-s + 49-s + 4.22·51-s − 6.57·53-s − 16.0·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.41·5-s + 0.377·7-s + 0.333·9-s − 1.52·11-s + 1.13·13-s + 0.819·15-s + 1.02·17-s + 1.16·19-s + 0.218·21-s − 0.208·23-s + 1.01·25-s + 0.192·27-s + 1.24·29-s + 0.398·31-s − 0.881·33-s + 0.536·35-s − 0.314·37-s + 0.652·39-s − 0.214·41-s + 0.517·43-s + 0.473·45-s + 0.848·47-s + 0.142·49-s + 0.591·51-s − 0.903·53-s − 2.16·55-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: \(0\)
Selberg data: \((2,\ 7728,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.992243708\)
\(L(\frac12)\) \(\approx\) \(3.992243708\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.17T + 5T^{2} \)
11 \( 1 + 5.06T + 11T^{2} \)
13 \( 1 - 4.07T + 13T^{2} \)
17 \( 1 - 4.22T + 17T^{2} \)
19 \( 1 - 5.06T + 19T^{2} \)
29 \( 1 - 6.68T + 29T^{2} \)
31 \( 1 - 2.22T + 31T^{2} \)
37 \( 1 + 1.91T + 37T^{2} \)
41 \( 1 + 1.37T + 41T^{2} \)
43 \( 1 - 3.39T + 43T^{2} \)
47 \( 1 - 5.81T + 47T^{2} \)
53 \( 1 + 6.57T + 53T^{2} \)
59 \( 1 - 5.67T + 59T^{2} \)
61 \( 1 + 14.4T + 61T^{2} \)
67 \( 1 + 13.8T + 67T^{2} \)
71 \( 1 - 2.95T + 71T^{2} \)
73 \( 1 + 2.02T + 73T^{2} \)
79 \( 1 + 7.14T + 79T^{2} \)
83 \( 1 - 12.7T + 83T^{2} \)
89 \( 1 + 17.0T + 89T^{2} \)
97 \( 1 - 2.08T + 97T^{2} \)
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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

−7.926813277136337257696804711769, −7.33050708722820324285181231296, −6.33507335886627301921727926713, −5.69567854999231425098470857562, −5.24519347285259693068286983220, −4.39317645585282890941837901331, −3.17974809303916313369342021820, −2.77608636978176645962163344564, −1.78647150633728923122749495521, −1.04912096756672502406369321281, 1.04912096756672502406369321281, 1.78647150633728923122749495521, 2.77608636978176645962163344564, 3.17974809303916313369342021820, 4.39317645585282890941837901331, 5.24519347285259693068286983220, 5.69567854999231425098470857562, 6.33507335886627301921727926713, 7.33050708722820324285181231296, 7.926813277136337257696804711769

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