Properties

Label 2-7728-1.1-c1-0-28
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 − 3.67·5-s − 7-s + 9-s + 6.24·11-s + 3.92·13-s − 3.67·15-s − 3.20·17-s − 5.20·19-s − 21-s − 23-s + 8.52·25-s + 27-s + 7.60·29-s − 2.15·31-s + 6.24·33-s + 3.67·35-s + 2.15·37-s + 3.92·39-s + 1.84·41-s + 4.57·43-s − 3.67·45-s − 5.35·47-s + 49-s − 3.20·51-s − 6.72·53-s − 22.9·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.64·5-s − 0.377·7-s + 0.333·9-s + 1.88·11-s + 1.08·13-s − 0.949·15-s − 0.776·17-s − 1.19·19-s − 0.218·21-s − 0.208·23-s + 1.70·25-s + 0.192·27-s + 1.41·29-s − 0.386·31-s + 1.08·33-s + 0.621·35-s + 0.354·37-s + 0.628·39-s + 0.288·41-s + 0.697·43-s − 0.548·45-s − 0.781·47-s + 0.142·49-s − 0.448·51-s − 0.923·53-s − 3.09·55-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

\(L(1)\) \(\approx\) \(1.831745600\)
\(L(\frac12)\) \(\approx\) \(1.831745600\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.67T + 5T^{2} \)
11 \( 1 - 6.24T + 11T^{2} \)
13 \( 1 - 3.92T + 13T^{2} \)
17 \( 1 + 3.20T + 17T^{2} \)
19 \( 1 + 5.20T + 19T^{2} \)
29 \( 1 - 7.60T + 29T^{2} \)
31 \( 1 + 2.15T + 31T^{2} \)
37 \( 1 - 2.15T + 37T^{2} \)
41 \( 1 - 1.84T + 41T^{2} \)
43 \( 1 - 4.57T + 43T^{2} \)
47 \( 1 + 5.35T + 47T^{2} \)
53 \( 1 + 6.72T + 53T^{2} \)
59 \( 1 - 12.7T + 59T^{2} \)
61 \( 1 + 11.5T + 61T^{2} \)
67 \( 1 + 8.72T + 67T^{2} \)
71 \( 1 + 10.0T + 71T^{2} \)
73 \( 1 - 11.2T + 73T^{2} \)
79 \( 1 - 5.29T + 79T^{2} \)
83 \( 1 - 8.24T + 83T^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 + 15.0T + 97T^{2} \)
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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

−8.024580538701320346578820748019, −7.14007549500221363358999757182, −6.56424004893800812947655026009, −6.12477993822057657532343594795, −4.56213073495298697879236417317, −4.18344328181042045884036831113, −3.66844199923267200525129268530, −2.95176216575052108638916451749, −1.70656951286178143510545561438, −0.67555425469213376190742891100, 0.67555425469213376190742891100, 1.70656951286178143510545561438, 2.95176216575052108638916451749, 3.66844199923267200525129268530, 4.18344328181042045884036831113, 4.56213073495298697879236417317, 6.12477993822057657532343594795, 6.56424004893800812947655026009, 7.14007549500221363358999757182, 8.024580538701320346578820748019

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