Properties

Label 2-7728-1.1-c1-0-32
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 − 2.59·5-s − 7-s + 9-s + 1.87·11-s + 0.721·13-s − 2.59·15-s + 5.31·17-s + 7.31·19-s − 21-s − 23-s + 1.72·25-s + 27-s − 5.31·29-s − 2.12·31-s + 1.87·33-s + 2.59·35-s − 6.49·37-s + 0.721·39-s + 4.12·41-s + 1.27·43-s − 2.59·45-s − 12.9·47-s + 49-s + 5.31·51-s + 10.3·53-s − 4.85·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.15·5-s − 0.377·7-s + 0.333·9-s + 0.564·11-s + 0.200·13-s − 0.669·15-s + 1.28·17-s + 1.67·19-s − 0.218·21-s − 0.208·23-s + 0.344·25-s + 0.192·27-s − 0.986·29-s − 0.382·31-s + 0.325·33-s + 0.438·35-s − 1.06·37-s + 0.115·39-s + 0.644·41-s + 0.194·43-s − 0.386·45-s − 1.88·47-s + 0.142·49-s + 0.744·51-s + 1.41·53-s − 0.654·55-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

\(L(1)\) \(\approx\) \(2.024490818\)
\(L(\frac12)\) \(\approx\) \(2.024490818\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.59T + 5T^{2} \)
11 \( 1 - 1.87T + 11T^{2} \)
13 \( 1 - 0.721T + 13T^{2} \)
17 \( 1 - 5.31T + 17T^{2} \)
19 \( 1 - 7.31T + 19T^{2} \)
29 \( 1 + 5.31T + 29T^{2} \)
31 \( 1 + 2.12T + 31T^{2} \)
37 \( 1 + 6.49T + 37T^{2} \)
41 \( 1 - 4.12T + 41T^{2} \)
43 \( 1 - 1.27T + 43T^{2} \)
47 \( 1 + 12.9T + 47T^{2} \)
53 \( 1 - 10.3T + 53T^{2} \)
59 \( 1 - 10.0T + 59T^{2} \)
61 \( 1 + 7.09T + 61T^{2} \)
67 \( 1 + 2.03T + 67T^{2} \)
71 \( 1 - 11.9T + 71T^{2} \)
73 \( 1 + 5.57T + 73T^{2} \)
79 \( 1 - 11.3T + 79T^{2} \)
83 \( 1 + 1.87T + 83T^{2} \)
89 \( 1 + 1.90T + 89T^{2} \)
97 \( 1 - 7.87T + 97T^{2} \)
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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

−7.73580960183714474992865438133, −7.41835104197357002040724287655, −6.68932833347238910136892598409, −5.69453574052191907269160742152, −5.06299242621250682555869722905, −3.93511147958218462646876969022, −3.59073246894269986626381581739, −2.99975756083796142834340276070, −1.71720138899966393676174568787, −0.71025124504326412618283983753, 0.71025124504326412618283983753, 1.71720138899966393676174568787, 2.99975756083796142834340276070, 3.59073246894269986626381581739, 3.93511147958218462646876969022, 5.06299242621250682555869722905, 5.69453574052191907269160742152, 6.68932833347238910136892598409, 7.41835104197357002040724287655, 7.73580960183714474992865438133

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