Properties

Label 2-1344-1.1-c1-0-21
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 $1$

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 − 8·23-s − 25-s − 27-s + 2·29-s + 4·33-s + 2·35-s + 10·37-s + 6·39-s − 6·41-s − 4·43-s + 2·45-s + 49-s − 2·51-s − 6·53-s − 8·55-s + 4·57-s + 4·59-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 − 1.66·23-s − 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.696·33-s + 0.338·35-s + 1.64·37-s + 0.960·39-s − 0.937·41-s − 0.609·43-s + 0.298·45-s + 1/7·49-s − 0.280·51-s − 0.824·53-s − 1.07·55-s + 0.529·57-s + 0.520·59-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: \(1\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−9.563538518206812440434518911703, −8.186632228453794414425905803942, −7.68501515018855362022465153557, −6.60807660052202573549483918899, −5.76536999919480153066315644301, −5.11740288034969242991482089170, −4.30167637849672643861137972376, −2.66792336770788585023270622114, −1.88351369279783613545674963411, 0, 1.88351369279783613545674963411, 2.66792336770788585023270622114, 4.30167637849672643861137972376, 5.11740288034969242991482089170, 5.76536999919480153066315644301, 6.60807660052202573549483918899, 7.68501515018855362022465153557, 8.186632228453794414425905803942, 9.563538518206812440434518911703

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