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Properties

Label 2-1368-456.125-c0-0-0
Degree $2$
Conductor $1368$
Sign $-0.616 + 0.787i$
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  + (0.258 + 0.965i)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 + (−0.258 − 0.965i)14-s + (0.500 − 0.866i)16-s + (−1.22 − 0.707i)17-s − i·19-s + (−0.366 − 1.36i)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  + (0.258 + 0.965i)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 + (−0.258 − 0.965i)14-s + (0.500 − 0.866i)16-s + (−1.22 − 0.707i)17-s − i·19-s + (−0.366 − 1.36i)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

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

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $-0.616 + 0.787i$
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: \((2,\ 1368,\ (\ :0),\ -0.616 + 0.787i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1756908154\)
\(L(\frac12)\) \(\approx\) \(0.1756908154\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
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

−9.882784184133641457742352957286, −9.533234213721485235323674037917, −8.555983191008471190945936669168, −7.78237561210190607488025676914, −6.78371802803151050172747499202, −6.55557498198546346617642804670, −5.07161371426719418394592672592, −4.87478181676076621292781495167, −3.42204358598977679980193540041, −2.57061950849077855512158796412, 0.12219064685438725820572970355, 2.24081155348599780146611428612, 2.83745098967168823708320291643, 4.02882418011929879375592175176, 4.82197095655708557180238278517, 5.88329675671193020379212721550, 6.53928712952409787716760704179, 8.025508302646692705649497722779, 8.406549146242335858124290213438, 9.712068531434004528013557192872

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