Properties

Label 2-1368-456.125-c0-0-3
Degree $2$
Conductor $1368$
Sign $0.787 + 0.616i$
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  + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s − 7-s + (0.707 − 0.707i)8-s + 1.41·11-s + (0.866 − 0.5i)13-s + (−0.965 + 0.258i)14-s + (0.500 − 0.866i)16-s + (−1.22 − 0.707i)17-s + i·19-s + (1.36 − 0.366i)22-s + (−1.22 + 0.707i)23-s + (0.5 + 0.866i)25-s + (0.707 − 0.707i)26-s + (−0.866 + 0.499i)28-s + (−0.707 − 1.22i)29-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s − 7-s + (0.707 − 0.707i)8-s + 1.41·11-s + (0.866 − 0.5i)13-s + (−0.965 + 0.258i)14-s + (0.500 − 0.866i)16-s + (−1.22 − 0.707i)17-s + i·19-s + (1.36 − 0.366i)22-s + (−1.22 + 0.707i)23-s + (0.5 + 0.866i)25-s + (0.707 − 0.707i)26-s + (−0.866 + 0.499i)28-s + (−0.707 − 1.22i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 + 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 + 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.787 + 0.616i$
Analytic conductor: \(0.682720\)
Root analytic conductor: \(0.826269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :0),\ 0.787 + 0.616i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.897892527\)
\(L(\frac12)\) \(\approx\) \(1.897892527\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
3 \( 1 \)
19 \( 1 - iT \)
good5 \( 1 + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + T + T^{2} \)
11 \( 1 - 1.41T + T^{2} \)
13 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 + iT - T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + 1.41T + T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.664540665487213913443580899123, −9.217321508354016527721705638601, −7.926428704350266602507879608877, −6.95422267772213781506828811564, −6.17262892353418037970524004743, −5.77271881217281302862697773064, −4.28778919219920853946433880112, −3.78394146839429030441905537050, −2.79484770197549617263948078139, −1.45157312927322497613592631426, 1.81321724579706628788823073695, 3.06827917098239451321384913778, 3.99360414659820788064266153440, 4.57289299100515247870989876542, 5.97132362634341421819158647402, 6.68458334463651323814165361439, 6.77100333495075315837130366968, 8.386934381627424615196995183814, 8.855028546787588983880095380898, 9.921007298064378938316095999726

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