Properties

Label 2-7728-1.1-c1-0-42
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 + 1.04·5-s + 7-s + 9-s − 0.180·11-s − 4.89·13-s + 1.04·15-s + 2.82·17-s + 0.180·19-s + 21-s − 23-s − 3.89·25-s + 27-s − 5.68·29-s + 0.825·31-s − 0.180·33-s + 1.04·35-s + 6.46·37-s − 4.89·39-s − 3.47·41-s − 0.125·43-s + 1.04·45-s + 12.0·47-s + 49-s + 2.82·51-s + 8.55·53-s − 0.189·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.469·5-s + 0.377·7-s + 0.333·9-s − 0.0545·11-s − 1.35·13-s + 0.270·15-s + 0.685·17-s + 0.0415·19-s + 0.218·21-s − 0.208·23-s − 0.779·25-s + 0.192·27-s − 1.05·29-s + 0.148·31-s − 0.0315·33-s + 0.177·35-s + 1.06·37-s − 0.784·39-s − 0.541·41-s − 0.0191·43-s + 0.156·45-s + 1.75·47-s + 0.142·49-s + 0.395·51-s + 1.17·53-s − 0.0255·55-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

\(L(1)\) \(\approx\) \(2.787555640\)
\(L(\frac12)\) \(\approx\) \(2.787555640\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.04T + 5T^{2} \)
11 \( 1 + 0.180T + 11T^{2} \)
13 \( 1 + 4.89T + 13T^{2} \)
17 \( 1 - 2.82T + 17T^{2} \)
19 \( 1 - 0.180T + 19T^{2} \)
29 \( 1 + 5.68T + 29T^{2} \)
31 \( 1 - 0.825T + 31T^{2} \)
37 \( 1 - 6.46T + 37T^{2} \)
41 \( 1 + 3.47T + 41T^{2} \)
43 \( 1 + 0.125T + 43T^{2} \)
47 \( 1 - 12.0T + 47T^{2} \)
53 \( 1 - 8.55T + 53T^{2} \)
59 \( 1 - 4.30T + 59T^{2} \)
61 \( 1 - 9.01T + 61T^{2} \)
67 \( 1 - 15.7T + 67T^{2} \)
71 \( 1 - 2.22T + 71T^{2} \)
73 \( 1 - 13.0T + 73T^{2} \)
79 \( 1 - 16.1T + 79T^{2} \)
83 \( 1 + 10.8T + 83T^{2} \)
89 \( 1 + 14.7T + 89T^{2} \)
97 \( 1 - 10.4T + 97T^{2} \)
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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

−7.75989847288795913968544135929, −7.38278922729787657632366565641, −6.57151541294488040492653965935, −5.58955568251429811498983489535, −5.19501825936628098460691835598, −4.21437178076774970850713983530, −3.55818599900669183758151734857, −2.41757717377684280771078897433, −2.09426787200334321054346579232, −0.801699269782003185872189277361, 0.801699269782003185872189277361, 2.09426787200334321054346579232, 2.41757717377684280771078897433, 3.55818599900669183758151734857, 4.21437178076774970850713983530, 5.19501825936628098460691835598, 5.58955568251429811498983489535, 6.57151541294488040492653965935, 7.38278922729787657632366565641, 7.75989847288795913968544135929

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