Properties

Label 2-7728-1.1-c1-0-47
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 + 2.75·5-s − 7-s + 9-s − 4.32·11-s + 1.57·13-s + 2.75·15-s + 0.819·17-s + 2.81·19-s − 21-s − 23-s + 2.57·25-s + 27-s − 0.819·29-s − 8.32·31-s − 4.32·33-s − 2.75·35-s + 8.68·37-s + 1.57·39-s + 10.3·41-s + 0.429·43-s + 2.75·45-s + 10.1·47-s + 49-s + 0.819·51-s − 7.39·53-s − 11.8·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.23·5-s − 0.377·7-s + 0.333·9-s − 1.30·11-s + 0.435·13-s + 0.710·15-s + 0.198·17-s + 0.646·19-s − 0.218·21-s − 0.208·23-s + 0.514·25-s + 0.192·27-s − 0.152·29-s − 1.49·31-s − 0.752·33-s − 0.465·35-s + 1.42·37-s + 0.251·39-s + 1.61·41-s + 0.0654·43-s + 0.410·45-s + 1.48·47-s + 0.142·49-s + 0.114·51-s − 1.01·53-s − 1.60·55-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

\(L(1)\) \(\approx\) \(3.070810146\)
\(L(\frac12)\) \(\approx\) \(3.070810146\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.75T + 5T^{2} \)
11 \( 1 + 4.32T + 11T^{2} \)
13 \( 1 - 1.57T + 13T^{2} \)
17 \( 1 - 0.819T + 17T^{2} \)
19 \( 1 - 2.81T + 19T^{2} \)
29 \( 1 + 0.819T + 29T^{2} \)
31 \( 1 + 8.32T + 31T^{2} \)
37 \( 1 - 8.68T + 37T^{2} \)
41 \( 1 - 10.3T + 41T^{2} \)
43 \( 1 - 0.429T + 43T^{2} \)
47 \( 1 - 10.1T + 47T^{2} \)
53 \( 1 + 7.39T + 53T^{2} \)
59 \( 1 - 6.39T + 59T^{2} \)
61 \( 1 - 13.4T + 61T^{2} \)
67 \( 1 - 1.60T + 67T^{2} \)
71 \( 1 - 2.06T + 71T^{2} \)
73 \( 1 + 13.4T + 73T^{2} \)
79 \( 1 - 6.81T + 79T^{2} \)
83 \( 1 - 4.32T + 83T^{2} \)
89 \( 1 - 7.93T + 89T^{2} \)
97 \( 1 - 1.67T + 97T^{2} \)
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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

−7.73232041428886798594865576631, −7.38215624669794160660434191612, −6.35364537974451138304594311103, −5.71632095607667363932008038772, −5.29724511215228100678591657205, −4.24421396934927834986293557814, −3.37160319587115260599304289965, −2.55050074414854664675864156041, −2.03652969213474175099983400934, −0.846769875939663071435630144965, 0.846769875939663071435630144965, 2.03652969213474175099983400934, 2.55050074414854664675864156041, 3.37160319587115260599304289965, 4.24421396934927834986293557814, 5.29724511215228100678591657205, 5.71632095607667363932008038772, 6.35364537974451138304594311103, 7.38215624669794160660434191612, 7.73232041428886798594865576631

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