Properties

Label 2-7728-1.1-c1-0-54
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 − 0.535·5-s + 7-s + 9-s + 2.85·11-s + 4.64·13-s − 0.535·15-s − 4.61·17-s + 2.85·19-s + 21-s + 23-s − 4.71·25-s + 27-s + 9.55·29-s + 3.22·31-s + 2.85·33-s − 0.535·35-s − 2.23·37-s + 4.64·39-s + 10.6·41-s − 0.687·43-s − 0.535·45-s − 2.01·47-s + 49-s − 4.61·51-s + 0.842·53-s − 1.52·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.239·5-s + 0.377·7-s + 0.333·9-s + 0.859·11-s + 1.28·13-s − 0.138·15-s − 1.11·17-s + 0.654·19-s + 0.218·21-s + 0.208·23-s − 0.942·25-s + 0.192·27-s + 1.77·29-s + 0.578·31-s + 0.496·33-s − 0.0905·35-s − 0.366·37-s + 0.743·39-s + 1.65·41-s − 0.104·43-s − 0.0798·45-s − 0.293·47-s + 0.142·49-s − 0.645·51-s + 0.115·53-s − 0.206·55-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

\(L(1)\) \(\approx\) \(3.074583224\)
\(L(\frac12)\) \(\approx\) \(3.074583224\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.535T + 5T^{2} \)
11 \( 1 - 2.85T + 11T^{2} \)
13 \( 1 - 4.64T + 13T^{2} \)
17 \( 1 + 4.61T + 17T^{2} \)
19 \( 1 - 2.85T + 19T^{2} \)
29 \( 1 - 9.55T + 29T^{2} \)
31 \( 1 - 3.22T + 31T^{2} \)
37 \( 1 + 2.23T + 37T^{2} \)
41 \( 1 - 10.6T + 41T^{2} \)
43 \( 1 + 0.687T + 43T^{2} \)
47 \( 1 + 2.01T + 47T^{2} \)
53 \( 1 - 0.842T + 53T^{2} \)
59 \( 1 + 7.87T + 59T^{2} \)
61 \( 1 + 5.76T + 61T^{2} \)
67 \( 1 - 4.55T + 67T^{2} \)
71 \( 1 + 7.39T + 71T^{2} \)
73 \( 1 + 14.9T + 73T^{2} \)
79 \( 1 - 1.54T + 79T^{2} \)
83 \( 1 - 14.4T + 83T^{2} \)
89 \( 1 + 0.327T + 89T^{2} \)
97 \( 1 - 11.8T + 97T^{2} \)
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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

−7.909427389102270426688951500925, −7.28499940611663422256004127075, −6.38731714648828348688002231292, −6.03991974029216745295965188510, −4.80223626675970517970383496383, −4.27174763590559876157112613027, −3.55562003804754493785107543530, −2.75299072604980445573256052700, −1.73222407040235872236063795864, −0.908682071045697644274408534152, 0.908682071045697644274408534152, 1.73222407040235872236063795864, 2.75299072604980445573256052700, 3.55562003804754493785107543530, 4.27174763590559876157112613027, 4.80223626675970517970383496383, 6.03991974029216745295965188510, 6.38731714648828348688002231292, 7.28499940611663422256004127075, 7.909427389102270426688951500925

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