Properties

Label 2-7728-1.1-c1-0-37
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 + 0.841·5-s − 7-s + 9-s + 4.45·11-s − 5.29·13-s + 0.841·15-s − 4.13·17-s − 2.13·19-s − 21-s − 23-s − 4.29·25-s + 27-s + 4.13·29-s + 0.451·31-s + 4.45·33-s − 0.841·35-s + 9.81·37-s − 5.29·39-s + 1.54·41-s + 7.29·43-s + 0.841·45-s − 11.2·47-s + 49-s − 4.13·51-s + 12.0·53-s + 3.74·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.376·5-s − 0.377·7-s + 0.333·9-s + 1.34·11-s − 1.46·13-s + 0.217·15-s − 1.00·17-s − 0.489·19-s − 0.218·21-s − 0.208·23-s − 0.858·25-s + 0.192·27-s + 0.767·29-s + 0.0810·31-s + 0.774·33-s − 0.142·35-s + 1.61·37-s − 0.847·39-s + 0.241·41-s + 1.11·43-s + 0.125·45-s − 1.63·47-s + 0.142·49-s − 0.578·51-s + 1.65·53-s + 0.504·55-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

\(L(1)\) \(\approx\) \(2.477961661\)
\(L(\frac12)\) \(\approx\) \(2.477961661\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.841T + 5T^{2} \)
11 \( 1 - 4.45T + 11T^{2} \)
13 \( 1 + 5.29T + 13T^{2} \)
17 \( 1 + 4.13T + 17T^{2} \)
19 \( 1 + 2.13T + 19T^{2} \)
29 \( 1 - 4.13T + 29T^{2} \)
31 \( 1 - 0.451T + 31T^{2} \)
37 \( 1 - 9.81T + 37T^{2} \)
41 \( 1 - 1.54T + 41T^{2} \)
43 \( 1 - 7.29T + 43T^{2} \)
47 \( 1 + 11.2T + 47T^{2} \)
53 \( 1 - 12.0T + 53T^{2} \)
59 \( 1 + 5.42T + 59T^{2} \)
61 \( 1 - 12.6T + 61T^{2} \)
67 \( 1 - 13.4T + 67T^{2} \)
71 \( 1 + 0.974T + 71T^{2} \)
73 \( 1 - 9.03T + 73T^{2} \)
79 \( 1 - 1.86T + 79T^{2} \)
83 \( 1 + 4.45T + 83T^{2} \)
89 \( 1 - 10.9T + 89T^{2} \)
97 \( 1 - 10.4T + 97T^{2} \)
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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

−7.86679658070491607975633487827, −7.11769474891368807835500575392, −6.52727030024421432156569661391, −5.97785695123838158487111004150, −4.87528907941776354008510640344, −4.26820836182968314401506966161, −3.57708541260758834980409212335, −2.45500633881841111008354546450, −2.08815254016717030275962245558, −0.74734194918063996957800009168, 0.74734194918063996957800009168, 2.08815254016717030275962245558, 2.45500633881841111008354546450, 3.57708541260758834980409212335, 4.26820836182968314401506966161, 4.87528907941776354008510640344, 5.97785695123838158487111004150, 6.52727030024421432156569661391, 7.11769474891368807835500575392, 7.86679658070491607975633487827

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