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Properties

Label 2-1344-168.149-c0-0-3
Degree $2$
Conductor $1344$
Sign $0.319 + 0.947i$
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  + (0.866 − 0.5i)3-s + (−0.866 − 0.5i)7-s + (0.499 − 0.866i)9-s − 1.73i·13-s + (−0.866 − 0.5i)19-s − 0.999·21-s + (0.5 + 0.866i)25-s − 0.999i·27-s + (0.866 + 1.5i)31-s + (1.5 + 0.866i)37-s + (−0.866 − 1.49i)39-s − i·43-s + (0.499 + 0.866i)49-s − 0.999·57-s + (−0.866 + 0.499i)63-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)3-s + (−0.866 − 0.5i)7-s + (0.499 − 0.866i)9-s − 1.73i·13-s + (−0.866 − 0.5i)19-s − 0.999·21-s + (0.5 + 0.866i)25-s − 0.999i·27-s + (0.866 + 1.5i)31-s + (1.5 + 0.866i)37-s + (−0.866 − 1.49i)39-s − i·43-s + (0.499 + 0.866i)49-s − 0.999·57-s + (−0.866 + 0.499i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.319 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.319 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.319 + 0.947i$
Analytic conductor: \(0.670743\)
Root analytic conductor: \(0.818989\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :0),\ 0.319 + 0.947i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.289214718\)
\(L(\frac12)\) \(\approx\) \(1.289214718\)
\(L(1)\) 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 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
good5 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + 1.73iT - T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + iT - T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + 2T + T^{2} \)
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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

−9.666568151584078414056768635480, −8.693059380700882561996749355591, −8.115344444575198194820695071719, −7.18635908409970359693114205415, −6.60814976326810454747600713874, −5.59382237170228901571769335358, −4.34183695681779948280261818766, −3.25162039990102499548465618776, −2.71509906724589432214580895289, −1.04121095484871988106285116300, 2.02820550402620440295195775208, 2.82862916578125221324399184367, 4.05419393233357474526461425241, 4.53173283628047613432420595293, 6.00031168300977893560136636422, 6.61720797693215607642056946776, 7.69528229476054894844123937891, 8.519700419940800743017882035225, 9.285730992491016336606917644137, 9.672759609801999461460872129168

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