Properties

Label 2-7728-1.1-c1-0-61
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 3.93·5-s + 7-s + 9-s − 2.69·11-s − 1.59·13-s + 3.93·15-s − 1.68·17-s − 2.69·19-s + 21-s + 23-s + 10.4·25-s + 27-s + 4.39·29-s + 8.29·31-s − 2.69·33-s + 3.93·35-s + 6.24·37-s − 1.59·39-s + 2.60·41-s − 10.2·43-s + 3.93·45-s + 3.07·47-s + 49-s − 1.68·51-s + 2.38·53-s − 10.6·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.75·5-s + 0.377·7-s + 0.333·9-s − 0.813·11-s − 0.441·13-s + 1.01·15-s − 0.407·17-s − 0.619·19-s + 0.218·21-s + 0.208·23-s + 2.09·25-s + 0.192·27-s + 0.816·29-s + 1.49·31-s − 0.469·33-s + 0.664·35-s + 1.02·37-s − 0.254·39-s + 0.406·41-s − 1.56·43-s + 0.586·45-s + 0.448·47-s + 0.142·49-s − 0.235·51-s + 0.328·53-s − 1.43·55-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

\(L(1)\) \(\approx\) \(3.813270954\)
\(L(\frac12)\) \(\approx\) \(3.813270954\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.93T + 5T^{2} \)
11 \( 1 + 2.69T + 11T^{2} \)
13 \( 1 + 1.59T + 13T^{2} \)
17 \( 1 + 1.68T + 17T^{2} \)
19 \( 1 + 2.69T + 19T^{2} \)
29 \( 1 - 4.39T + 29T^{2} \)
31 \( 1 - 8.29T + 31T^{2} \)
37 \( 1 - 6.24T + 37T^{2} \)
41 \( 1 - 2.60T + 41T^{2} \)
43 \( 1 + 10.2T + 43T^{2} \)
47 \( 1 - 3.07T + 47T^{2} \)
53 \( 1 - 2.38T + 53T^{2} \)
59 \( 1 + 1.63T + 59T^{2} \)
61 \( 1 + 1.29T + 61T^{2} \)
67 \( 1 + 11.0T + 67T^{2} \)
71 \( 1 - 7.70T + 71T^{2} \)
73 \( 1 - 9.36T + 73T^{2} \)
79 \( 1 - 7.54T + 79T^{2} \)
83 \( 1 + 2.16T + 83T^{2} \)
89 \( 1 - 5.84T + 89T^{2} \)
97 \( 1 - 7.64T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.990300906422095995097169176002, −7.13681562379016719633513654462, −6.36282130527423968405966198847, −5.90982071378046167217577919912, −4.85045896170326576720347646699, −4.68053138656830208834788849609, −3.25504298334978458034956427917, −2.42055117258442146790351930221, −2.09487490023616780842556930740, −0.966685214681881007894364630013, 0.966685214681881007894364630013, 2.09487490023616780842556930740, 2.42055117258442146790351930221, 3.25504298334978458034956427917, 4.68053138656830208834788849609, 4.85045896170326576720347646699, 5.90982071378046167217577919912, 6.36282130527423968405966198847, 7.13681562379016719633513654462, 7.990300906422095995097169176002

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