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Properties

Label 2-1368-1368.1019-c0-0-1
Degree $2$
Conductor $1368$
Sign $0.699 + 0.714i$
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)12-s + 16-s + (−1.5 + 0.866i)17-s + (−0.499 − 0.866i)18-s − 19-s + (0.5 − 0.866i)24-s + (0.5 + 0.866i)25-s − 0.999·27-s + 32-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)12-s + 16-s + (−1.5 + 0.866i)17-s + (−0.499 − 0.866i)18-s − 19-s + (0.5 − 0.866i)24-s + (0.5 + 0.866i)25-s − 0.999·27-s + 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 + 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 + 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.699 + 0.714i$
Analytic conductor: \(0.682720\)
Root analytic conductor: \(0.826269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (1019, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :0),\ 0.699 + 0.714i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.247133729\)
\(L(\frac12)\) \(\approx\) \(2.247133729\)
\(L(1)\) 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 \( 1 + T \)
good5 \( 1 + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
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 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + 1.73iT - T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + T^{2} \)
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 \( 1 - T^{2} \)
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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

−9.645283325981158141300416177522, −8.603574794593597963786900524688, −8.022572272044741461334092658294, −6.89272122826084991012093978262, −6.56649810600448946036840716606, −5.61742885590298475158585292665, −4.47637269167217473955573615766, −3.60316751573377103670795345528, −2.52308499792742952572811717848, −1.67247916555918633457811860114, 2.19004075749737114168407515878, 2.90370896257695148726069880142, 4.13935682905790601537629219185, 4.54941426375695327588876820909, 5.51862397367057283459929811540, 6.49605851361768181737660137189, 7.31059235504777561464214283150, 8.375799794996604349316021392341, 9.035382161108632739797915950459, 10.05738317743343404393516529408

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