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Properties

Label 2-1368-1368.1075-c0-0-2
Degree $2$
Conductor $1368$
Sign $0.612 - 0.790i$
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  + 2-s + (−0.5 + 0.866i)3-s + 4-s + (−0.5 + 0.866i)6-s + 8-s + (−0.499 − 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.5 + 0.866i)12-s + 16-s + (0.5 − 0.866i)17-s + (−0.499 − 0.866i)18-s + (−0.5 + 0.866i)19-s + (0.5 + 0.866i)22-s + (−0.5 + 0.866i)24-s + (−0.5 + 0.866i)25-s + ⋯
L(s)  = 1  + 2-s + (−0.5 + 0.866i)3-s + 4-s + (−0.5 + 0.866i)6-s + 8-s + (−0.499 − 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.5 + 0.866i)12-s + 16-s + (0.5 − 0.866i)17-s + (−0.499 − 0.866i)18-s + (−0.5 + 0.866i)19-s + (0.5 + 0.866i)22-s + (−0.5 + 0.866i)24-s + (−0.5 + 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.612 - 0.790i$
Analytic conductor: \(0.682720\)
Root analytic conductor: \(0.826269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (1075, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :0),\ 0.612 - 0.790i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.761875808\)
\(L(\frac12)\) \(\approx\) \(1.761875808\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.959161577990900682258233736298, −9.448101999875661682021079428351, −8.200496083450288031370306366030, −7.16388383461423760372204156393, −6.44396145670729245783316950852, −5.49763570087490325524422019751, −4.89004857543489799639344870133, −3.96685342899043908544354573930, −3.26824092855577981618522693483, −1.81663886755901956018562637406, 1.35917532374469826631182757309, 2.54001284021414971578418490426, 3.61783465249980083822586076150, 4.71444878540678821472544809858, 5.61333046101326014519586695911, 6.38262839318434213840635797776, 6.81681449248100472005855428732, 7.984628726720632745776299382721, 8.469573491655852700780784530475, 9.933395296642072710366356307055

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