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Properties

Label 2-2808-312.269-c0-0-1
Degree $2$
Conductor $2808$
Sign $0.505 - 0.862i$
Analytic cond. $1.40137$
Root an. cond. $1.18379$
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 + (−0.707 − 0.707i)8-s − i·13-s + (0.500 − 0.866i)16-s + (1.22 − 0.707i)17-s + (0.866 − 0.5i)19-s + (1.22 + 0.707i)23-s − 25-s + (0.965 − 0.258i)26-s + (−0.707 + 1.22i)29-s + 31-s + (0.965 + 0.258i)32-s + (1 + 0.999i)34-s + (0.866 + 0.5i)37-s + (0.707 + 0.707i)38-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (−0.707 − 0.707i)8-s − i·13-s + (0.500 − 0.866i)16-s + (1.22 − 0.707i)17-s + (0.866 − 0.5i)19-s + (1.22 + 0.707i)23-s − 25-s + (0.965 − 0.258i)26-s + (−0.707 + 1.22i)29-s + 31-s + (0.965 + 0.258i)32-s + (1 + 0.999i)34-s + (0.866 + 0.5i)37-s + (0.707 + 0.707i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2808\)    =    \(2^{3} \cdot 3^{3} \cdot 13\)
Sign: $0.505 - 0.862i$
Analytic conductor: \(1.40137\)
Root analytic conductor: \(1.18379\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2808} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2808,\ (\ :0),\ 0.505 - 0.862i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.314340673\)
\(L(\frac12)\) \(\approx\) \(1.314340673\)
\(L(1)\) 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 \( 1 \)
13 \( 1 + iT \)
good5 \( 1 + T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
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 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 - 1.41iT - T^{2} \)
53 \( 1 + 1.41T + T^{2} \)
59 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + T + T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 - 1.41T + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.043954276954705850761609115274, −8.126610036159923439735812622739, −7.48013232792543107858593455729, −7.03492636684555279933071996959, −5.87194487215050139791334771531, −5.38131177546474375873999268693, −4.70102707161127343196357447801, −3.45334501527198821090906291092, −2.97034609599529079434561117133, −1.02026587231404982824013366085, 1.14164366727025013572954536858, 2.16819207018575213623791876507, 3.20678841280352177865620280326, 3.98911361597986412411728505255, 4.76175134433404551008169185210, 5.70135467983343362932083160192, 6.31666602022070866052393981457, 7.51214474842431024241220448213, 8.220105038874777094821570728660, 9.058607302642620275128338481339

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