Properties

Label 2-26928-1.1-c1-0-11
Degree $2$
Conductor $26928$
Sign $1$
Analytic cond. $215.021$
Root an. cond. $14.6635$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·7-s − 11-s − 2·13-s + 17-s + 4·19-s + 6·23-s − 5·25-s + 4·29-s + 2·31-s − 4·37-s + 2·41-s + 4·43-s − 3·49-s − 2·53-s + 4·59-s − 12·67-s + 2·71-s + 2·73-s − 2·77-s + 14·79-s + 12·83-s − 6·89-s − 4·91-s − 2·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  + 0.755·7-s − 0.301·11-s − 0.554·13-s + 0.242·17-s + 0.917·19-s + 1.25·23-s − 25-s + 0.742·29-s + 0.359·31-s − 0.657·37-s + 0.312·41-s + 0.609·43-s − 3/7·49-s − 0.274·53-s + 0.520·59-s − 1.46·67-s + 0.237·71-s + 0.234·73-s − 0.227·77-s + 1.57·79-s + 1.31·83-s − 0.635·89-s − 0.419·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(26928\)    =    \(2^{4} \cdot 3^{2} \cdot 11 \cdot 17\)
Sign: $1$
Analytic conductor: \(215.021\)
Root analytic conductor: \(14.6635\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 26928,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.506081977\)
\(L(\frac12)\) \(\approx\) \(2.506081977\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 + T \)
17 \( 1 - T \)
good5 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−15.08089406873507, −14.90847024193202, −14.11332625028846, −13.77140431722442, −13.24765125784138, −12.42910585142795, −12.13027752856958, −11.47730930933530, −11.01384399848667, −10.43263798826051, −9.797568809787115, −9.337970975033255, −8.643552567932641, −8.045974159160313, −7.540129381807662, −7.072832395817686, −6.281292778870555, −5.587520900837342, −4.975274403979943, −4.619612438530123, −3.680230805875750, −3.008583752348152, −2.297292786448140, −1.465268358968674, −0.6455380053171409, 0.6455380053171409, 1.465268358968674, 2.297292786448140, 3.008583752348152, 3.680230805875750, 4.619612438530123, 4.975274403979943, 5.587520900837342, 6.281292778870555, 7.072832395817686, 7.540129381807662, 8.045974159160313, 8.643552567932641, 9.337970975033255, 9.797568809787115, 10.43263798826051, 11.01384399848667, 11.47730930933530, 12.13027752856958, 12.42910585142795, 13.24765125784138, 13.77140431722442, 14.11332625028846, 14.90847024193202, 15.08089406873507

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