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Properties

Label 2-1368-1368.1211-c0-0-1
Degree $2$
Conductor $1368$
Sign $-0.226 + 0.973i$
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  + (−0.766 − 0.642i)2-s + (0.766 − 0.642i)3-s + (0.173 + 0.984i)4-s − 6-s + (0.500 − 0.866i)8-s + (0.173 − 0.984i)9-s − 1.28i·11-s + (0.766 + 0.642i)12-s + (−0.939 + 0.342i)16-s + (−0.766 + 0.642i)18-s + (0.939 + 0.342i)19-s + (−0.826 + 0.984i)22-s + (−0.173 − 0.984i)24-s + (−0.766 + 0.642i)25-s + (−0.500 − 0.866i)27-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (0.766 − 0.642i)3-s + (0.173 + 0.984i)4-s − 6-s + (0.500 − 0.866i)8-s + (0.173 − 0.984i)9-s − 1.28i·11-s + (0.766 + 0.642i)12-s + (−0.939 + 0.342i)16-s + (−0.766 + 0.642i)18-s + (0.939 + 0.342i)19-s + (−0.826 + 0.984i)22-s + (−0.173 − 0.984i)24-s + (−0.766 + 0.642i)25-s + (−0.500 − 0.866i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.226 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.226 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $-0.226 + 0.973i$
Analytic conductor: \(0.682720\)
Root analytic conductor: \(0.826269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (1211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :0),\ -0.226 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9585332948\)
\(L(\frac12)\) \(\approx\) \(0.9585332948\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 + 0.642i)T \)
3 \( 1 + (-0.766 + 0.642i)T \)
19 \( 1 + (-0.939 - 0.342i)T \)
good5 \( 1 + (0.766 - 0.642i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + 1.28iT - T^{2} \)
13 \( 1 + (0.766 + 0.642i)T^{2} \)
17 \( 1 + (-0.766 + 0.642i)T^{2} \)
23 \( 1 + (-0.939 + 0.342i)T^{2} \)
29 \( 1 + (0.939 - 0.342i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \)
43 \( 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2} \)
47 \( 1 + (-0.939 + 0.342i)T^{2} \)
53 \( 1 + (-0.173 + 0.984i)T^{2} \)
59 \( 1 + (-0.326 + 1.85i)T + (-0.939 - 0.342i)T^{2} \)
61 \( 1 + (-0.766 - 0.642i)T^{2} \)
67 \( 1 + (-1.26 - 1.50i)T + (-0.173 + 0.984i)T^{2} \)
71 \( 1 + (-0.173 - 0.984i)T^{2} \)
73 \( 1 + (0.266 - 1.50i)T + (-0.939 - 0.342i)T^{2} \)
79 \( 1 + (0.766 - 0.642i)T^{2} \)
83 \( 1 + (1.11 + 0.642i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.173 - 0.984i)T + (-0.939 + 0.342i)T^{2} \)
97 \( 1 + (0.439 - 0.524i)T + (-0.173 - 0.984i)T^{2} \)
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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

−9.504232172310861372996557971130, −8.687885997972855761758094181397, −8.146265544346743594159365484017, −7.41397602684084898733487468552, −6.58470914981122727453355298837, −5.50380403265297795744480525536, −3.83600505515921740858644808050, −3.27167770139591081961690382207, −2.21471740895542666896887142672, −1.02245587372791056146818541246, 1.73300115883335327550641907956, 2.80456000473685649504763568100, 4.25567832106121680556507376856, 4.94546464258787497687906931143, 5.96214555124808662776067605922, 7.11197217530985486495510172186, 7.65921174517806939282727435932, 8.406339849792768639411888900261, 9.375032247803913506563612246855, 9.686253497770736014306261957995

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