Properties

Label 2-7728-1.1-c1-0-1
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 + 0.0840·5-s − 7-s + 9-s − 5.90·11-s − 4.15·13-s − 0.0840·15-s − 2.07·17-s − 8.23·19-s + 21-s + 23-s − 4.99·25-s − 27-s + 6.65·29-s − 4.07·31-s + 5.90·33-s − 0.0840·35-s − 6.65·37-s + 4.15·39-s − 6.65·41-s − 11.9·43-s + 0.0840·45-s + 4.58·47-s + 49-s + 2.07·51-s + 7.00·53-s − 0.496·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.0375·5-s − 0.377·7-s + 0.333·9-s − 1.78·11-s − 1.15·13-s − 0.0216·15-s − 0.502·17-s − 1.88·19-s + 0.218·21-s + 0.208·23-s − 0.998·25-s − 0.192·27-s + 1.23·29-s − 0.731·31-s + 1.02·33-s − 0.0142·35-s − 1.09·37-s + 0.665·39-s − 1.03·41-s − 1.82·43-s + 0.0125·45-s + 0.669·47-s + 0.142·49-s + 0.289·51-s + 0.961·53-s − 0.0669·55-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

\(L(1)\) \(\approx\) \(0.2407211725\)
\(L(\frac12)\) \(\approx\) \(0.2407211725\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.0840T + 5T^{2} \)
11 \( 1 + 5.90T + 11T^{2} \)
13 \( 1 + 4.15T + 13T^{2} \)
17 \( 1 + 2.07T + 17T^{2} \)
19 \( 1 + 8.23T + 19T^{2} \)
29 \( 1 - 6.65T + 29T^{2} \)
31 \( 1 + 4.07T + 31T^{2} \)
37 \( 1 + 6.65T + 37T^{2} \)
41 \( 1 + 6.65T + 41T^{2} \)
43 \( 1 + 11.9T + 43T^{2} \)
47 \( 1 - 4.58T + 47T^{2} \)
53 \( 1 - 7.00T + 53T^{2} \)
59 \( 1 + 12.4T + 59T^{2} \)
61 \( 1 - 8.74T + 61T^{2} \)
67 \( 1 - 12.6T + 67T^{2} \)
71 \( 1 - 10.9T + 71T^{2} \)
73 \( 1 + 8.07T + 73T^{2} \)
79 \( 1 - 5.90T + 79T^{2} \)
83 \( 1 + 7.90T + 83T^{2} \)
89 \( 1 - 11.0T + 89T^{2} \)
97 \( 1 - 16.3T + 97T^{2} \)
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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

−7.942644499222171897418433303964, −6.93720289368250155868072014249, −6.65453763022112012233631491529, −5.65159432559142224814978064214, −5.09233277278588084153785471729, −4.52492767741483929568266531894, −3.54090092107996657745162497137, −2.50377414338157838836436987277, −1.98686803841350165897613031885, −0.23168702351168136629563322453, 0.23168702351168136629563322453, 1.98686803841350165897613031885, 2.50377414338157838836436987277, 3.54090092107996657745162497137, 4.52492767741483929568266531894, 5.09233277278588084153785471729, 5.65159432559142224814978064214, 6.65453763022112012233631491529, 6.93720289368250155868072014249, 7.942644499222171897418433303964

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