Properties

Label 2-7728-1.1-c1-0-115
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 + 0.703·5-s − 7-s + 9-s − 0.606·11-s − 1.30·13-s + 0.703·15-s − 3.67·17-s + 0.606·19-s − 21-s + 23-s − 4.50·25-s + 27-s + 5.63·29-s − 1.86·31-s − 0.606·33-s − 0.703·35-s − 2.80·37-s − 1.30·39-s − 0.329·41-s + 2.99·43-s + 0.703·45-s + 4.19·47-s + 49-s − 3.67·51-s − 10.0·53-s − 0.426·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.314·5-s − 0.377·7-s + 0.333·9-s − 0.182·11-s − 0.363·13-s + 0.181·15-s − 0.890·17-s + 0.139·19-s − 0.218·21-s + 0.208·23-s − 0.900·25-s + 0.192·27-s + 1.04·29-s − 0.335·31-s − 0.105·33-s − 0.118·35-s − 0.460·37-s − 0.209·39-s − 0.0513·41-s + 0.456·43-s + 0.104·45-s + 0.611·47-s + 0.142·49-s − 0.514·51-s − 1.37·53-s − 0.0575·55-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 \( 1 - 0.703T + 5T^{2} \)
11 \( 1 + 0.606T + 11T^{2} \)
13 \( 1 + 1.30T + 13T^{2} \)
17 \( 1 + 3.67T + 17T^{2} \)
19 \( 1 - 0.606T + 19T^{2} \)
29 \( 1 - 5.63T + 29T^{2} \)
31 \( 1 + 1.86T + 31T^{2} \)
37 \( 1 + 2.80T + 37T^{2} \)
41 \( 1 + 0.329T + 41T^{2} \)
43 \( 1 - 2.99T + 43T^{2} \)
47 \( 1 - 4.19T + 47T^{2} \)
53 \( 1 + 10.0T + 53T^{2} \)
59 \( 1 - 9.73T + 59T^{2} \)
61 \( 1 + 13.9T + 61T^{2} \)
67 \( 1 + 8.66T + 67T^{2} \)
71 \( 1 - 3.86T + 71T^{2} \)
73 \( 1 - 5.82T + 73T^{2} \)
79 \( 1 - 1.86T + 79T^{2} \)
83 \( 1 + 13.5T + 83T^{2} \)
89 \( 1 + 14.9T + 89T^{2} \)
97 \( 1 - 5.82T + 97T^{2} \)
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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

−7.53507968815591016922365090813, −6.85941438649516440666965323919, −6.22032836045820273334671888339, −5.42268859005913717433808459723, −4.60221215370727771804884439155, −3.91143905421051222283407749834, −2.97593440308285178608419789218, −2.37065408272788500202491933588, −1.42337354704132020736888631440, 0, 1.42337354704132020736888631440, 2.37065408272788500202491933588, 2.97593440308285178608419789218, 3.91143905421051222283407749834, 4.60221215370727771804884439155, 5.42268859005913717433808459723, 6.22032836045820273334671888339, 6.85941438649516440666965323919, 7.53507968815591016922365090813

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