Properties

Label 2-1344-1.1-c1-0-8
Degree $2$
Conductor $1344$
Sign $1$
Analytic cond. $10.7318$
Root an. cond. $3.27595$
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 + 2·5-s − 7-s + 9-s + 4·11-s + 6·13-s − 2·15-s − 2·17-s − 4·19-s + 21-s − 4·23-s − 25-s − 27-s + 2·29-s + 8·31-s − 4·33-s − 2·35-s + 10·37-s − 6·39-s − 2·41-s − 8·43-s + 2·45-s + 49-s + 2·51-s + 10·53-s + 8·55-s + 4·57-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s + 1.66·13-s − 0.516·15-s − 0.485·17-s − 0.917·19-s + 0.218·21-s − 0.834·23-s − 1/5·25-s − 0.192·27-s + 0.371·29-s + 1.43·31-s − 0.696·33-s − 0.338·35-s + 1.64·37-s − 0.960·39-s − 0.312·41-s − 1.21·43-s + 0.298·45-s + 1/7·49-s + 0.280·51-s + 1.37·53-s + 1.07·55-s + 0.529·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $1$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.744581748\)
\(L(\frac12)\) \(\approx\) \(1.744581748\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
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

−9.747193101497493749604763487169, −8.839713456987639065173433643950, −8.190173500954779681525780731187, −6.66339990127808429560142628788, −6.35837082667327210170937090345, −5.74243037997109166467332435643, −4.43219841019389924830737127106, −3.69109001567993476839628496737, −2.19510980306732737447197047942, −1.06346104514984424021583213227, 1.06346104514984424021583213227, 2.19510980306732737447197047942, 3.69109001567993476839628496737, 4.43219841019389924830737127106, 5.74243037997109166467332435643, 6.35837082667327210170937090345, 6.66339990127808429560142628788, 8.190173500954779681525780731187, 8.839713456987639065173433643950, 9.747193101497493749604763487169

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