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Properties

Label 2-1368-1368.619-c0-0-2
Degree $2$
Conductor $1368$
Sign $0.884 + 0.466i$
Analytic cond. $0.682720$
Root an. cond. $0.826269$
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)2-s + 3-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)6-s + 0.999·8-s + 9-s + (−1 + 1.73i)11-s + (−0.499 + 0.866i)12-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)18-s + 19-s + 1.99·22-s + 0.999·24-s + 25-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + 3-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)6-s + 0.999·8-s + 9-s + (−1 + 1.73i)11-s + (−0.499 + 0.866i)12-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)18-s + 19-s + 1.99·22-s + 0.999·24-s + 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.884 + 0.466i$
Analytic conductor: \(0.682720\)
Root analytic conductor: \(0.826269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (619, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :0),\ 0.884 + 0.466i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.145459403\)
\(L(\frac12)\) \(\approx\) \(1.145459403\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 - T \)
19 \( 1 - T \)
good5 \( 1 - T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.917453027297138826619245402409, −9.094589215196535197811536862598, −8.209570800648944948529434388791, −7.45203618342923373147519079661, −7.05118191042878853508342862901, −5.08793541727939659086968727682, −4.50366156497486134580940444674, −3.25330572035040421517049450868, −2.58378302575382613485139916506, −1.52505675615170011341675871588, 1.23469924277622365506106348909, 2.82281648285893078076516027249, 3.73521511468091973331326751073, 5.02791190707556246349661063140, 5.78476422841888052494694663692, 6.74558511018088958535175811911, 7.72070293778092032737757934644, 8.192851587799892192257133027893, 8.822890583496170453687575602297, 9.576653552210071286089572194378

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