Properties

Label 2-1824-1.1-c1-0-12
Degree $2$
Conductor $1824$
Sign $1$
Analytic cond. $14.5647$
Root an. cond. $3.81637$
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  + 3-s + 9-s − 2·17-s + 19-s + 6·23-s − 5·25-s + 27-s + 10·29-s + 10·31-s + 8·37-s + 6·41-s − 4·43-s − 6·47-s − 7·49-s − 2·51-s − 6·53-s + 57-s + 12·59-s − 10·61-s + 8·67-s + 6·69-s + 8·71-s − 2·73-s − 5·75-s + 6·79-s + 81-s + 12·83-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 0.485·17-s + 0.229·19-s + 1.25·23-s − 25-s + 0.192·27-s + 1.85·29-s + 1.79·31-s + 1.31·37-s + 0.937·41-s − 0.609·43-s − 0.875·47-s − 49-s − 0.280·51-s − 0.824·53-s + 0.132·57-s + 1.56·59-s − 1.28·61-s + 0.977·67-s + 0.722·69-s + 0.949·71-s − 0.234·73-s − 0.577·75-s + 0.675·79-s + 1/9·81-s + 1.31·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1824\)    =    \(2^{5} \cdot 3 \cdot 19\)
Sign: $1$
Analytic conductor: \(14.5647\)
Root analytic conductor: \(3.81637\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1824,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.228524341\)
\(L(\frac12)\) \(\approx\) \(2.228524341\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
19 \( 1 - T \)
good5 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
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

−9.355356742299644543676367616318, −8.296496221262657576898415819468, −7.964470068878401325040700572312, −6.81088904583270023450399436591, −6.27599751315138271157133868156, −5.00303538629533549971548364042, −4.34266529251789130500314196237, −3.19208687181736396542673311061, −2.42350504986374656226702469922, −1.03905423431556020745421685829, 1.03905423431556020745421685829, 2.42350504986374656226702469922, 3.19208687181736396542673311061, 4.34266529251789130500314196237, 5.00303538629533549971548364042, 6.27599751315138271157133868156, 6.81088904583270023450399436591, 7.964470068878401325040700572312, 8.296496221262657576898415819468, 9.355356742299644543676367616318

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