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Properties

Label 2-1344-168.83-c0-0-3
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  + 3-s + 2i·5-s − i·7-s + 9-s + 2i·15-s − i·21-s − 3·25-s + 27-s + 2·35-s + 2i·45-s − 49-s − 2·59-s − i·63-s − 3·75-s − 2i·79-s + ⋯
L(s)  = 1  + 3-s + 2i·5-s − i·7-s + 9-s + 2i·15-s − i·21-s − 3·25-s + 27-s + 2·35-s + 2i·45-s − 49-s − 2·59-s − i·63-s − 3·75-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

\(L(\frac{1}{2})\) \(\approx\) \(1.512124793\)
\(L(\frac12)\) \(\approx\) \(1.512124793\)
\(L(1)\) 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 \( 1 - T \)
7 \( 1 + iT \)
good5 \( 1 - 2iT - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
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 \( 1 + 2T + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + 2iT - T^{2} \)
83 \( 1 - 2T + T^{2} \)
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

−10.02434591454822508523044007990, −9.254872699624438923274803529712, −8.006874265078102742595785631489, −7.46879546943563674588330691429, −6.85889690125891550436222230047, −6.12599192989108251760440954702, −4.49251423262248452317565761389, −3.55582733445626620624457324083, −2.99681241204836855571021607027, −1.90439843024767588587168139043, 1.36498208514417184712220884500, 2.36980558671651843623068820600, 3.69594011156376324014248735374, 4.66384702907679232095241813168, 5.29917338346508283270828660573, 6.31697904491644397344879327658, 7.73106681065734215678369462028, 8.220247017225351282970206479300, 9.075397515750723303655878742632, 9.208584020646936779053932831057

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