Properties

Label 2-7728-1.1-c1-0-53
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 + 0.351·5-s + 7-s + 9-s − 3.41·11-s + 6.57·13-s + 0.351·15-s + 7.98·17-s − 3.41·19-s + 21-s + 23-s − 4.87·25-s + 27-s + 2.98·29-s − 1.19·31-s − 3.41·33-s + 0.351·35-s + 2.74·37-s + 6.57·39-s − 11.4·41-s + 2.84·43-s + 0.351·45-s + 7.45·47-s + 49-s + 7.98·51-s + 3.01·53-s − 1.19·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.156·5-s + 0.377·7-s + 0.333·9-s − 1.02·11-s + 1.82·13-s + 0.0906·15-s + 1.93·17-s − 0.782·19-s + 0.218·21-s + 0.208·23-s − 0.975·25-s + 0.192·27-s + 0.555·29-s − 0.214·31-s − 0.593·33-s + 0.0593·35-s + 0.451·37-s + 1.05·39-s − 1.78·41-s + 0.433·43-s + 0.0523·45-s + 1.08·47-s + 0.142·49-s + 1.11·51-s + 0.414·53-s − 0.161·55-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

\(L(1)\) \(\approx\) \(3.127593481\)
\(L(\frac12)\) \(\approx\) \(3.127593481\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.351T + 5T^{2} \)
11 \( 1 + 3.41T + 11T^{2} \)
13 \( 1 - 6.57T + 13T^{2} \)
17 \( 1 - 7.98T + 17T^{2} \)
19 \( 1 + 3.41T + 19T^{2} \)
29 \( 1 - 2.98T + 29T^{2} \)
31 \( 1 + 1.19T + 31T^{2} \)
37 \( 1 - 2.74T + 37T^{2} \)
41 \( 1 + 11.4T + 41T^{2} \)
43 \( 1 - 2.84T + 43T^{2} \)
47 \( 1 - 7.45T + 47T^{2} \)
53 \( 1 - 3.01T + 53T^{2} \)
59 \( 1 - 4.07T + 59T^{2} \)
61 \( 1 - 9.00T + 61T^{2} \)
67 \( 1 - 7.91T + 67T^{2} \)
71 \( 1 - 9.21T + 71T^{2} \)
73 \( 1 + 4.47T + 73T^{2} \)
79 \( 1 + 9.28T + 79T^{2} \)
83 \( 1 + 9.73T + 83T^{2} \)
89 \( 1 - 14.5T + 89T^{2} \)
97 \( 1 + 1.09T + 97T^{2} \)
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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

−8.076534924897013328029215614603, −7.33775468858580664686352987544, −6.45506444979280573991554422814, −5.67447690030307035642849887416, −5.22717137134642686582413997350, −4.08045016674930086524201433496, −3.56071387109352125623172186186, −2.72094837910258982318021663860, −1.78659491045548428330746173993, −0.902678039315383940439711508014, 0.902678039315383940439711508014, 1.78659491045548428330746173993, 2.72094837910258982318021663860, 3.56071387109352125623172186186, 4.08045016674930086524201433496, 5.22717137134642686582413997350, 5.67447690030307035642849887416, 6.45506444979280573991554422814, 7.33775468858580664686352987544, 8.076534924897013328029215614603

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