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Properties

Label 2-1344-168.59-c0-0-7
Degree $2$
Conductor $1344$
Sign $-0.925 + 0.378i$
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  + (0.5 − 0.866i)3-s + (−0.866 + 0.5i)5-s + (−0.866 − 0.5i)7-s + (−0.499 − 0.866i)9-s + (−1.5 − 0.866i)11-s + 0.999i·15-s + (−0.866 + 0.499i)21-s − 0.999·27-s − 1.73·29-s + (0.866 − 1.5i)31-s + (−1.5 + 0.866i)33-s + 0.999·35-s + (0.866 + 0.499i)45-s + (0.499 + 0.866i)49-s + (0.866 − 1.5i)53-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s + (−0.866 + 0.5i)5-s + (−0.866 − 0.5i)7-s + (−0.499 − 0.866i)9-s + (−1.5 − 0.866i)11-s + 0.999i·15-s + (−0.866 + 0.499i)21-s − 0.999·27-s − 1.73·29-s + (0.866 − 1.5i)31-s + (−1.5 + 0.866i)33-s + 0.999·35-s + (0.866 + 0.499i)45-s + (0.499 + 0.866i)49-s + (0.866 − 1.5i)53-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.925 + 0.378i$
Analytic conductor: \(0.670743\)
Root analytic conductor: \(0.818989\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (479, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :0),\ -0.925 + 0.378i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5158393017\)
\(L(\frac12)\) \(\approx\) \(0.5158393017\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
good5 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + 1.73T + T^{2} \)
31 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T + T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + 1.73iT - T^{2} \)
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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

−9.417094974176847499983829380371, −8.366209257881868980350677900255, −7.74231102829829740786454244172, −7.23736832863158853143003445640, −6.31377679835593194596590557652, −5.47896598582697144793438417959, −3.89312199972765814276043041693, −3.26051838614669978343674285327, −2.35379096703489875231038907299, −0.37657584773981582957058983616, 2.33257629487174856787672796317, 3.21901305721781383322176111221, 4.18771551070668449845377448889, 4.99888388467824141213865775686, 5.76029336523597300922322639434, 7.16772972479712609318525820594, 7.86406001574949865046259458014, 8.634365304197086713631549719127, 9.312558076973444673722185297035, 10.14801138929974697275414327446

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