Properties

Label 2-7728-1.1-c1-0-15
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 − 2.70·5-s − 7-s + 9-s + 4·11-s − 0.701·13-s + 2.70·15-s + 4·17-s − 7.40·19-s + 21-s − 23-s + 2.29·25-s − 27-s + 6.70·29-s + 2·31-s − 4·33-s + 2.70·35-s + 10.7·37-s + 0.701·39-s − 6.70·41-s − 4.70·43-s − 2.70·45-s − 8.10·47-s + 49-s − 4·51-s + 3.40·53-s − 10.8·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.20·5-s − 0.377·7-s + 0.333·9-s + 1.20·11-s − 0.194·13-s + 0.697·15-s + 0.970·17-s − 1.69·19-s + 0.218·21-s − 0.208·23-s + 0.459·25-s − 0.192·27-s + 1.24·29-s + 0.359·31-s − 0.696·33-s + 0.456·35-s + 1.75·37-s + 0.112·39-s − 1.04·41-s − 0.716·43-s − 0.402·45-s − 1.18·47-s + 0.142·49-s − 0.560·51-s + 0.467·53-s − 1.45·55-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

\(L(1)\) \(\approx\) \(0.9404224530\)
\(L(\frac12)\) \(\approx\) \(0.9404224530\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.70T + 5T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 + 0.701T + 13T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 + 7.40T + 19T^{2} \)
29 \( 1 - 6.70T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 - 10.7T + 37T^{2} \)
41 \( 1 + 6.70T + 41T^{2} \)
43 \( 1 + 4.70T + 43T^{2} \)
47 \( 1 + 8.10T + 47T^{2} \)
53 \( 1 - 3.40T + 53T^{2} \)
59 \( 1 + 5.40T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 10.8T + 67T^{2} \)
71 \( 1 + 5.40T + 71T^{2} \)
73 \( 1 + 11.4T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 0.596T + 83T^{2} \)
89 \( 1 - 1.40T + 89T^{2} \)
97 \( 1 - 12.7T + 97T^{2} \)
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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

−7.930028048545582102393908123795, −7.05665042589200869385539318578, −6.46875349338535535528497497749, −6.00680814245760931563076776175, −4.85575657100300267313996839362, −4.28617306465315880877684609901, −3.71028616607640851484528745142, −2.85613960146247795396373140031, −1.56572353180311439580611930246, −0.51521887791342157925015831321, 0.51521887791342157925015831321, 1.56572353180311439580611930246, 2.85613960146247795396373140031, 3.71028616607640851484528745142, 4.28617306465315880877684609901, 4.85575657100300267313996839362, 6.00680814245760931563076776175, 6.46875349338535535528497497749, 7.05665042589200869385539318578, 7.930028048545582102393908123795

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