Properties

Label 2-7728-1.1-c1-0-118
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 − 1.08·5-s − 7-s + 9-s + 4.07·11-s + 5.16·13-s − 1.08·15-s − 2.56·17-s − 4.07·19-s − 21-s + 23-s − 3.82·25-s + 27-s − 10.6·29-s − 6.55·31-s + 4.07·33-s + 1.08·35-s − 3.90·37-s + 5.16·39-s − 1.43·41-s − 12.8·43-s − 1.08·45-s + 9.98·47-s + 49-s − 2.56·51-s + 9.15·53-s − 4.42·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.485·5-s − 0.377·7-s + 0.333·9-s + 1.22·11-s + 1.43·13-s − 0.280·15-s − 0.622·17-s − 0.935·19-s − 0.218·21-s + 0.208·23-s − 0.764·25-s + 0.192·27-s − 1.97·29-s − 1.17·31-s + 0.709·33-s + 0.183·35-s − 0.642·37-s + 0.826·39-s − 0.224·41-s − 1.95·43-s − 0.161·45-s + 1.45·47-s + 0.142·49-s − 0.359·51-s + 1.25·53-s − 0.597·55-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: \(1\)
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 \( 1 + 1.08T + 5T^{2} \)
11 \( 1 - 4.07T + 11T^{2} \)
13 \( 1 - 5.16T + 13T^{2} \)
17 \( 1 + 2.56T + 17T^{2} \)
19 \( 1 + 4.07T + 19T^{2} \)
29 \( 1 + 10.6T + 29T^{2} \)
31 \( 1 + 6.55T + 31T^{2} \)
37 \( 1 + 3.90T + 37T^{2} \)
41 \( 1 + 1.43T + 41T^{2} \)
43 \( 1 + 12.8T + 43T^{2} \)
47 \( 1 - 9.98T + 47T^{2} \)
53 \( 1 - 9.15T + 53T^{2} \)
59 \( 1 + 3.63T + 59T^{2} \)
61 \( 1 - 1.90T + 61T^{2} \)
67 \( 1 - 8.28T + 67T^{2} \)
71 \( 1 + 14.1T + 71T^{2} \)
73 \( 1 + 4.64T + 73T^{2} \)
79 \( 1 - 6.55T + 79T^{2} \)
83 \( 1 + 8.86T + 83T^{2} \)
89 \( 1 + 10.6T + 89T^{2} \)
97 \( 1 + 4.64T + 97T^{2} \)
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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

−7.39838587328432027034031959994, −6.95630084807477388861839937501, −6.18443197021728849331216585962, −5.57187220425158857894489281674, −4.33833982914250271876476406274, −3.76255089402131327106942529173, −3.46663249532224866881665705833, −2.12878789994202413107149587426, −1.43848535218019959271765267053, 0, 1.43848535218019959271765267053, 2.12878789994202413107149587426, 3.46663249532224866881665705833, 3.76255089402131327106942529173, 4.33833982914250271876476406274, 5.57187220425158857894489281674, 6.18443197021728849331216585962, 6.95630084807477388861839937501, 7.39838587328432027034031959994

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