Properties

Label 2-7728-1.1-c1-0-129
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 + 1.47·5-s + 7-s + 9-s − 4.58·11-s + 3.11·13-s + 1.47·15-s − 6.58·17-s + 6.81·19-s + 21-s − 23-s − 2.83·25-s + 27-s − 7.30·29-s − 1.64·31-s − 4.58·33-s + 1.47·35-s − 8.10·37-s + 3.11·39-s − 9.87·41-s − 1.11·43-s + 1.47·45-s − 10.2·47-s + 49-s − 6.58·51-s − 5.47·53-s − 6.75·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.658·5-s + 0.377·7-s + 0.333·9-s − 1.38·11-s + 0.863·13-s + 0.380·15-s − 1.59·17-s + 1.56·19-s + 0.218·21-s − 0.208·23-s − 0.566·25-s + 0.192·27-s − 1.35·29-s − 0.294·31-s − 0.798·33-s + 0.248·35-s − 1.33·37-s + 0.498·39-s − 1.54·41-s − 0.170·43-s + 0.219·45-s − 1.49·47-s + 0.142·49-s − 0.922·51-s − 0.751·53-s − 0.911·55-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 \( 1 - 1.47T + 5T^{2} \)
11 \( 1 + 4.58T + 11T^{2} \)
13 \( 1 - 3.11T + 13T^{2} \)
17 \( 1 + 6.58T + 17T^{2} \)
19 \( 1 - 6.81T + 19T^{2} \)
29 \( 1 + 7.30T + 29T^{2} \)
31 \( 1 + 1.64T + 31T^{2} \)
37 \( 1 + 8.10T + 37T^{2} \)
41 \( 1 + 9.87T + 41T^{2} \)
43 \( 1 + 1.11T + 43T^{2} \)
47 \( 1 + 10.2T + 47T^{2} \)
53 \( 1 + 5.47T + 53T^{2} \)
59 \( 1 + 12.9T + 59T^{2} \)
61 \( 1 + 7.49T + 61T^{2} \)
67 \( 1 - 8.64T + 67T^{2} \)
71 \( 1 - 6.96T + 71T^{2} \)
73 \( 1 - 15.0T + 73T^{2} \)
79 \( 1 - 2.69T + 79T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 - 11.9T + 89T^{2} \)
97 \( 1 - 7.04T + 97T^{2} \)
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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

−7.69983836402075293965766816075, −6.85897995574540721311179824843, −6.14581331503310050383597523245, −5.24344766672783959241678497733, −4.92234037204368515572225867887, −3.72799314032337939568877598408, −3.13786577984137574191803727846, −2.09120734402972391108175203718, −1.63345868575511824481165799233, 0, 1.63345868575511824481165799233, 2.09120734402972391108175203718, 3.13786577984137574191803727846, 3.72799314032337939568877598408, 4.92234037204368515572225867887, 5.24344766672783959241678497733, 6.14581331503310050383597523245, 6.85897995574540721311179824843, 7.69983836402075293965766816075

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