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Properties

Label 2-1368-1368.403-c0-0-0
Degree $2$
Conductor $1368$
Sign $0.845 + 0.533i$
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 + 1.53·11-s + (0.766 − 0.642i)12-s + (−0.939 − 0.342i)16-s + (−1.87 − 0.684i)17-s + (0.766 + 0.642i)18-s + (−0.939 + 0.342i)19-s + (1.17 − 0.984i)22-s + (0.173 − 0.984i)24-s + (0.766 + 0.642i)25-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 + 1.53·11-s + (0.766 − 0.642i)12-s + (−0.939 − 0.342i)16-s + (−1.87 − 0.684i)17-s + (0.766 + 0.642i)18-s + (−0.939 + 0.342i)19-s + (1.17 − 0.984i)22-s + (0.173 − 0.984i)24-s + (0.766 + 0.642i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.845 + 0.533i$
Analytic conductor: \(0.682720\)
Root analytic conductor: \(0.826269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :0),\ 0.845 + 0.533i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.064215402\)
\(L(\frac12)\) \(\approx\) \(2.064215402\)
\(L(1)\) 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 \( 1 - 1.53T + T^{2} \)
13 \( 1 + (-0.766 + 0.642i)T^{2} \)
17 \( 1 + (1.87 + 0.684i)T + (0.766 + 0.642i)T^{2} \)
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 \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \)
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 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
97 \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \)
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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

−9.580071833823956378718674006394, −9.174575790285775515551785736367, −8.454724931319345814773569659766, −7.00271240200450086613372220157, −6.44418933587013520245286181624, −5.17913680831518238405860598478, −4.32842368214115029869908058915, −3.80071021914470413803482622418, −2.68605637015677730009040488767, −1.72341843891266410999098124536, 1.83561561708543029341117683936, 2.87097021543909185438652324061, 4.04202721541876520505888493464, 4.50057603988980234923567831222, 6.12686579241117019209457073015, 6.58278747222411848489497225308, 7.14026877214549245101218761494, 8.367291773466754564995851403469, 8.690624188989585039733513401028, 9.447379619969073850355361960321

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