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Properties

Label 2-1344-168.83-c0-0-7
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

\(L(\frac{1}{2})\) \(\approx\) \(0.8157877663\)
\(L(\frac12)\) \(\approx\) \(0.8157877663\)
\(L(1)\) 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

−9.457818700670686484362034482537, −8.633981357776708861677121797666, −7.57730701215358861084635402820, −7.13227714042248432464393531804, −6.57273757678871610247577869121, −5.20353217517582207709577092423, −4.57784368867396479056914895631, −3.08120214068228317677806424379, −2.23105734622805977135489512021, −0.65597087790900529738681733472, 2.19120588441234514115597611919, 3.08569452244465344015797761851, 4.20600723611712189144672787205, 5.16939002276792563186168539567, 5.64798320921760840170288700174, 6.77837805955879714663078744850, 7.916189395212009002813251088120, 8.565175954157963719945820308180, 9.545000781749769850760113908114, 9.902396192078867387683212222100

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