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Properties

Label 2-1368-1368.1091-c0-0-1
Degree $2$
Conductor $1368$
Sign $-0.0765 + 0.997i$
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 + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s − 0.999·6-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (1.5 + 0.866i)11-s + (0.499 + 0.866i)12-s + (−0.5 − 0.866i)16-s + (1.5 + 0.866i)17-s + (−0.499 + 0.866i)18-s + (0.5 − 0.866i)19-s − 1.73i·22-s + (0.499 − 0.866i)24-s − 25-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s − 0.999·6-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (1.5 + 0.866i)11-s + (0.499 + 0.866i)12-s + (−0.5 − 0.866i)16-s + (1.5 + 0.866i)17-s + (−0.499 + 0.866i)18-s + (0.5 − 0.866i)19-s − 1.73i·22-s + (0.499 − 0.866i)24-s − 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0765 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0765 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $-0.0765 + 0.997i$
Analytic conductor: \(0.682720\)
Root analytic conductor: \(0.826269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (1091, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :0),\ -0.0765 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.038640137\)
\(L(\frac12)\) \(\approx\) \(1.038640137\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 + 2T + T^{2} \)
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 \( 1 - 2T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
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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

−9.691436649731093004354316564194, −8.755898085737955640536920359841, −8.167677057350766313972447672209, −7.23881186745261518698054045612, −6.67774911319977830811908147441, −5.35733960171460311550567419590, −3.96405014548236946409448763123, −3.36367202777032325278742053583, −2.05351319272735566573960121163, −1.26001695613468266332237377801, 1.44007819683684671557996598702, 3.27412113949562425648995345187, 4.01138017801110516411969887790, 5.17340891428238780603104114864, 5.81667675292754270818685920830, 6.77702984096318369222910684008, 7.84952708259084775549217524152, 8.329777552555637362625282839459, 9.283277366879567426371252237900, 9.700968481574518812859086126834

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