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Properties

Label 2-1344-168.83-c0-0-4
Degree $2$
Conductor $1344$
Sign $0.707 + 0.707i$
Analytic cond. $0.670743$
Root an. cond. $0.818989$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·3-s + i·7-s − 9-s + 2·13-s − 2i·19-s + 21-s + 25-s + i·27-s − 2i·39-s − 49-s − 2·57-s − 2·61-s − i·63-s − i·75-s + 2i·79-s + ⋯
L(s)  = 1  − i·3-s + i·7-s − 9-s + 2·13-s − 2i·19-s + 21-s + 25-s + i·27-s − 2i·39-s − 49-s − 2·57-s − 2·61-s − i·63-s − i·75-s + 2i·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(0.670743\)
Root analytic conductor: \(0.818989\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (671, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :0),\ 0.707 + 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.128234316\)
\(L(\frac12)\) \(\approx\) \(1.128234316\)
\(L(1)\) 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 \( 1 + iT \)
7 \( 1 - iT \)
good5 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - 2T + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + 2iT - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + 2T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - 2iT - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.354954356029106958045375089875, −8.741660049457947189601230705677, −8.292130196846028609239823856448, −7.13977745847292536294915545202, −6.42183063986854377059349482217, −5.77339543976341204223818388542, −4.79153249099958305568443874700, −3.32011807519595132741865495142, −2.47495016111105270558228759483, −1.20403815503451087888988435018, 1.40486403518532242249004867753, 3.28439258559214431625703184604, 3.80971667961244198549523359263, 4.65190651024797205426139835890, 5.82341410171504914367181872037, 6.38460940670539142674041580352, 7.67004482517405927059557654228, 8.402172298659248132592523459432, 9.102455760899311584670151755378, 10.11200839042751556043772798304

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