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Properties

Label 2-1344-168.131-c0-0-5
Degree $2$
Conductor $1344$
Sign $-0.378 + 0.925i$
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 − 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 − 0.5i)39-s + 1.73·43-s + (0.499 + 0.866i)49-s + 0.999·57-s + (1 − 1.73i)61-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)3-s + (−0.866 − 0.5i)7-s + (0.499 − 0.866i)9-s − 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 − 0.5i)39-s + 1.73·43-s + (0.499 + 0.866i)49-s + 0.999·57-s + (1 − 1.73i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.378 + 0.925i$
Analytic conductor: \(0.670743\)
Root analytic conductor: \(0.818989\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (1055, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :0),\ -0.378 + 0.925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3375674861\)
\(L(\frac12)\) \(\approx\) \(0.3375674861\)
\(L(1)\) 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 \( 1 + T + T^{2} \)
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 \( 1 - 1.73T + T^{2} \)
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 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
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

−9.596052278234456036938968768839, −9.115218075735689881029570725447, −7.75771974515419393837473677475, −6.95578508763412800923466119043, −6.24532231749518591682194779552, −5.40157792134676826059930968489, −4.38755021438058114502277638558, −3.71182975263978808558764606077, −2.33112047000576801605323796811, −0.29940503505185706885603444074, 1.72272788128356616326467380485, 2.89466446918807618209886096158, 4.17832744246761997907180527899, 5.30886579331559694634466792919, 5.85311639275698056361156541600, 6.86986185145877035196134362474, 7.31039260978814106425547395575, 8.461279472645633319220572967085, 9.300235506989734615074913472275, 10.22364654532849960918831321550

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