Properties

Label 2-7728-1.1-c1-0-19
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 − 1.36·5-s − 7-s + 9-s − 5.49·11-s + 6.48·13-s + 1.36·15-s + 7.12·17-s + 3.85·19-s + 21-s + 23-s − 3.13·25-s − 27-s − 6.11·29-s + 5.12·31-s + 5.49·33-s + 1.36·35-s + 6.11·37-s − 6.48·39-s + 6.11·41-s − 10.1·43-s − 1.36·45-s + 1.00·47-s + 49-s − 7.12·51-s − 9.70·53-s + 7.51·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.610·5-s − 0.377·7-s + 0.333·9-s − 1.65·11-s + 1.79·13-s + 0.352·15-s + 1.72·17-s + 0.883·19-s + 0.218·21-s + 0.208·23-s − 0.626·25-s − 0.192·27-s − 1.13·29-s + 0.919·31-s + 0.957·33-s + 0.230·35-s + 1.00·37-s − 1.03·39-s + 0.954·41-s − 1.54·43-s − 0.203·45-s + 0.147·47-s + 0.142·49-s − 0.997·51-s − 1.33·53-s + 1.01·55-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

\(L(1)\) \(\approx\) \(1.222370236\)
\(L(\frac12)\) \(\approx\) \(1.222370236\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.36T + 5T^{2} \)
11 \( 1 + 5.49T + 11T^{2} \)
13 \( 1 - 6.48T + 13T^{2} \)
17 \( 1 - 7.12T + 17T^{2} \)
19 \( 1 - 3.85T + 19T^{2} \)
29 \( 1 + 6.11T + 29T^{2} \)
31 \( 1 - 5.12T + 31T^{2} \)
37 \( 1 - 6.11T + 37T^{2} \)
41 \( 1 - 6.11T + 41T^{2} \)
43 \( 1 + 10.1T + 43T^{2} \)
47 \( 1 - 1.00T + 47T^{2} \)
53 \( 1 + 9.70T + 53T^{2} \)
59 \( 1 + 0.754T + 59T^{2} \)
61 \( 1 + 5.47T + 61T^{2} \)
67 \( 1 + 1.97T + 67T^{2} \)
71 \( 1 + 6.20T + 71T^{2} \)
73 \( 1 + 4.76T + 73T^{2} \)
79 \( 1 - 5.49T + 79T^{2} \)
83 \( 1 + 1.61T + 83T^{2} \)
89 \( 1 + 14.8T + 89T^{2} \)
97 \( 1 - 15.2T + 97T^{2} \)
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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

−7.79300416263993892496749312811, −7.39928349542061477586015264465, −6.26254935103106955795790687928, −5.78724992845178542693617825152, −5.22514149932849615997625082874, −4.30642653584781865656582889669, −3.41451130176187605649505031326, −2.99346176723810677507822285932, −1.54695494321202444580371487863, −0.59457269870187268847450115063, 0.59457269870187268847450115063, 1.54695494321202444580371487863, 2.99346176723810677507822285932, 3.41451130176187605649505031326, 4.30642653584781865656582889669, 5.22514149932849615997625082874, 5.78724992845178542693617825152, 6.26254935103106955795790687928, 7.39928349542061477586015264465, 7.79300416263993892496749312811

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