Properties

Label 2-7728-1.1-c1-0-11
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 − 2.43·5-s + 7-s + 9-s − 2.13·11-s + 6.02·13-s + 2.43·15-s − 6.45·17-s − 7.14·19-s − 21-s − 23-s + 0.923·25-s − 27-s + 0.407·29-s + 8.45·31-s + 2.13·33-s − 2.43·35-s + 0.407·37-s − 6.02·39-s − 0.901·41-s − 3.60·43-s − 2.43·45-s + 4.46·47-s + 49-s + 6.45·51-s − 1.89·53-s + 5.19·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.08·5-s + 0.377·7-s + 0.333·9-s − 0.644·11-s + 1.67·13-s + 0.628·15-s − 1.56·17-s − 1.63·19-s − 0.218·21-s − 0.208·23-s + 0.184·25-s − 0.192·27-s + 0.0756·29-s + 1.51·31-s + 0.371·33-s − 0.411·35-s + 0.0669·37-s − 0.964·39-s − 0.140·41-s − 0.549·43-s − 0.362·45-s + 0.651·47-s + 0.142·49-s + 0.903·51-s − 0.259·53-s + 0.701·55-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

\(L(1)\) \(\approx\) \(0.8258488182\)
\(L(\frac12)\) \(\approx\) \(0.8258488182\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.43T + 5T^{2} \)
11 \( 1 + 2.13T + 11T^{2} \)
13 \( 1 - 6.02T + 13T^{2} \)
17 \( 1 + 6.45T + 17T^{2} \)
19 \( 1 + 7.14T + 19T^{2} \)
29 \( 1 - 0.407T + 29T^{2} \)
31 \( 1 - 8.45T + 31T^{2} \)
37 \( 1 - 0.407T + 37T^{2} \)
41 \( 1 + 0.901T + 41T^{2} \)
43 \( 1 + 3.60T + 43T^{2} \)
47 \( 1 - 4.46T + 47T^{2} \)
53 \( 1 + 1.89T + 53T^{2} \)
59 \( 1 + 3.89T + 59T^{2} \)
61 \( 1 - 5.06T + 61T^{2} \)
67 \( 1 + 8.53T + 67T^{2} \)
71 \( 1 + 1.75T + 71T^{2} \)
73 \( 1 - 8.51T + 73T^{2} \)
79 \( 1 + 8.04T + 79T^{2} \)
83 \( 1 + 3.58T + 83T^{2} \)
89 \( 1 - 4.16T + 89T^{2} \)
97 \( 1 + 1.69T + 97T^{2} \)
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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

−8.029892628144342788810440131381, −7.10308355230874130007880730486, −6.40896447539782181220039324434, −5.96670499560694133735403651734, −4.86131442037234731543833063058, −4.29553716448384950962482479097, −3.81946266627414014183777116107, −2.68387575015293223157036173136, −1.68351427661747331627061605918, −0.46553247525756662928056469181, 0.46553247525756662928056469181, 1.68351427661747331627061605918, 2.68387575015293223157036173136, 3.81946266627414014183777116107, 4.29553716448384950962482479097, 4.86131442037234731543833063058, 5.96670499560694133735403651734, 6.40896447539782181220039324434, 7.10308355230874130007880730486, 8.029892628144342788810440131381

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