Properties

Label 2-7728-1.1-c1-0-91
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 0.462·5-s + 7-s + 9-s − 1.32·11-s + 1.78·13-s + 0.462·15-s − 2.39·17-s + 7.04·19-s − 21-s + 23-s − 4.78·25-s − 27-s − 3.32·29-s − 7.04·31-s + 1.32·33-s − 0.462·35-s − 7.32·37-s − 1.78·39-s + 1.04·41-s + 1.13·43-s − 0.462·45-s + 11.0·47-s + 49-s + 2.39·51-s + 0.591·53-s + 0.612·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.206·5-s + 0.377·7-s + 0.333·9-s − 0.399·11-s + 0.495·13-s + 0.119·15-s − 0.581·17-s + 1.61·19-s − 0.218·21-s + 0.208·23-s − 0.957·25-s − 0.192·27-s − 0.617·29-s − 1.26·31-s + 0.230·33-s − 0.0781·35-s − 1.20·37-s − 0.285·39-s + 0.163·41-s + 0.173·43-s − 0.0689·45-s + 1.60·47-s + 0.142·49-s + 0.335·51-s + 0.0812·53-s + 0.0825·55-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7728,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.462T + 5T^{2} \)
11 \( 1 + 1.32T + 11T^{2} \)
13 \( 1 - 1.78T + 13T^{2} \)
17 \( 1 + 2.39T + 17T^{2} \)
19 \( 1 - 7.04T + 19T^{2} \)
29 \( 1 + 3.32T + 29T^{2} \)
31 \( 1 + 7.04T + 31T^{2} \)
37 \( 1 + 7.32T + 37T^{2} \)
41 \( 1 - 1.04T + 41T^{2} \)
43 \( 1 - 1.13T + 43T^{2} \)
47 \( 1 - 11.0T + 47T^{2} \)
53 \( 1 - 0.591T + 53T^{2} \)
59 \( 1 - 6.46T + 59T^{2} \)
61 \( 1 - 0.860T + 61T^{2} \)
67 \( 1 + 14.7T + 67T^{2} \)
71 \( 1 - 2.73T + 71T^{2} \)
73 \( 1 + 15.3T + 73T^{2} \)
79 \( 1 - 4.39T + 79T^{2} \)
83 \( 1 + 12.7T + 83T^{2} \)
89 \( 1 - 7.37T + 89T^{2} \)
97 \( 1 + 8.11T + 97T^{2} \)
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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

−7.26552733976399072432180783877, −7.13375951367453045568039632272, −5.74717356347840943021861397705, −5.68139952668086965078713808687, −4.74510633141001600890146618113, −3.96435730573117127229746463035, −3.23732779323222932598160179286, −2.11439243022250460751533825598, −1.21079396516726511565422059844, 0, 1.21079396516726511565422059844, 2.11439243022250460751533825598, 3.23732779323222932598160179286, 3.96435730573117127229746463035, 4.74510633141001600890146618113, 5.68139952668086965078713808687, 5.74717356347840943021861397705, 7.13375951367453045568039632272, 7.26552733976399072432180783877

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