Properties

Label 2-7728-1.1-c1-0-5
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 − 3.21·5-s − 7-s + 9-s − 3.79·11-s − 6.57·13-s − 3.21·15-s + 0.454·17-s − 1.54·19-s − 21-s − 23-s + 5.33·25-s + 27-s − 3.36·29-s − 4.88·31-s − 3.79·33-s + 3.21·35-s + 4.88·37-s − 6.57·39-s − 0.883·41-s − 5.00·43-s − 3.21·45-s − 4.42·47-s + 49-s + 0.454·51-s + 0.123·53-s + 12.1·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.43·5-s − 0.377·7-s + 0.333·9-s − 1.14·11-s − 1.82·13-s − 0.829·15-s + 0.110·17-s − 0.354·19-s − 0.218·21-s − 0.208·23-s + 1.06·25-s + 0.192·27-s − 0.624·29-s − 0.877·31-s − 0.660·33-s + 0.543·35-s + 0.802·37-s − 1.05·39-s − 0.137·41-s − 0.763·43-s − 0.479·45-s − 0.645·47-s + 0.142·49-s + 0.0636·51-s + 0.0170·53-s + 1.64·55-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: \(0\)
Selberg data: \((2,\ 7728,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5202704672\)
\(L(\frac12)\) \(\approx\) \(0.5202704672\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.21T + 5T^{2} \)
11 \( 1 + 3.79T + 11T^{2} \)
13 \( 1 + 6.57T + 13T^{2} \)
17 \( 1 - 0.454T + 17T^{2} \)
19 \( 1 + 1.54T + 19T^{2} \)
29 \( 1 + 3.36T + 29T^{2} \)
31 \( 1 + 4.88T + 31T^{2} \)
37 \( 1 - 4.88T + 37T^{2} \)
41 \( 1 + 0.883T + 41T^{2} \)
43 \( 1 + 5.00T + 43T^{2} \)
47 \( 1 + 4.42T + 47T^{2} \)
53 \( 1 - 0.123T + 53T^{2} \)
59 \( 1 - 5.87T + 59T^{2} \)
61 \( 1 + 8.33T + 61T^{2} \)
67 \( 1 + 1.87T + 67T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 + 5.13T + 73T^{2} \)
79 \( 1 + 11.1T + 79T^{2} \)
83 \( 1 + 1.79T + 83T^{2} \)
89 \( 1 - 0.150T + 89T^{2} \)
97 \( 1 - 9.61T + 97T^{2} \)
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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

−7.82713227222982077365012894613, −7.36475129666248187744644476244, −6.82096482588298673040405124894, −5.64957256346011092664150617399, −4.87572433866845635562140879823, −4.28674609710486782548160537392, −3.44637116159750082615850509267, −2.81030178753131464128305461195, −2.01389359606571696671366565934, −0.32420786596393055999396332525, 0.32420786596393055999396332525, 2.01389359606571696671366565934, 2.81030178753131464128305461195, 3.44637116159750082615850509267, 4.28674609710486782548160537392, 4.87572433866845635562140879823, 5.64957256346011092664150617399, 6.82096482588298673040405124894, 7.36475129666248187744644476244, 7.82713227222982077365012894613

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