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Properties

Label 1-2668-2668.1319-r0-0-0
Degree $1$
Conductor $2668$
Sign $0.938 - 0.345i$
Analytic cond. $12.3901$
Root an. cond. $12.3901$
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  + (0.396 + 0.917i)3-s + (−0.996 + 0.0815i)5-s + (−0.0203 + 0.999i)7-s + (−0.685 + 0.728i)9-s + (−0.925 − 0.377i)11-s + (0.452 + 0.891i)13-s + (−0.470 − 0.882i)15-s + (−0.540 − 0.841i)17-s + (0.122 − 0.992i)19-s + (−0.925 + 0.377i)21-s + (0.986 − 0.162i)25-s + (−0.940 − 0.339i)27-s + (−0.806 − 0.591i)31-s + (−0.0203 − 0.999i)33-s + (−0.0611 − 0.998i)35-s + ⋯
L(s)  = 1  + (0.396 + 0.917i)3-s + (−0.996 + 0.0815i)5-s + (−0.0203 + 0.999i)7-s + (−0.685 + 0.728i)9-s + (−0.925 − 0.377i)11-s + (0.452 + 0.891i)13-s + (−0.470 − 0.882i)15-s + (−0.540 − 0.841i)17-s + (0.122 − 0.992i)19-s + (−0.925 + 0.377i)21-s + (0.986 − 0.162i)25-s + (−0.940 − 0.339i)27-s + (−0.806 − 0.591i)31-s + (−0.0203 − 0.999i)33-s + (−0.0611 − 0.998i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2668 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2668 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2668\)    =    \(2^{2} \cdot 23 \cdot 29\)
Sign: $0.938 - 0.345i$
Analytic conductor: \(12.3901\)
Root analytic conductor: \(12.3901\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2668} (1319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2668,\ (0:\ ),\ 0.938 - 0.345i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7170676019 - 0.1279658235i\)
\(L(\frac12)\) \(\approx\) \(0.7170676019 - 0.1279658235i\)
\(L(1)\) \(\approx\) \(0.7774753383 + 0.2755317147i\)
\(L(1)\) \(\approx\) \(0.7774753383 + 0.2755317147i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good3 \( 1 + (0.396 + 0.917i)T \)
5 \( 1 + (-0.996 + 0.0815i)T \)
7 \( 1 + (-0.0203 + 0.999i)T \)
11 \( 1 + (-0.925 - 0.377i)T \)
13 \( 1 + (0.452 + 0.891i)T \)
17 \( 1 + (-0.540 - 0.841i)T \)
19 \( 1 + (0.122 - 0.992i)T \)
31 \( 1 + (-0.806 - 0.591i)T \)
37 \( 1 + (0.728 + 0.685i)T \)
41 \( 1 + (-0.989 - 0.142i)T \)
43 \( 1 + (0.806 - 0.591i)T \)
47 \( 1 + (-0.781 - 0.623i)T \)
53 \( 1 + (0.794 - 0.607i)T \)
59 \( 1 + (-0.415 + 0.909i)T \)
61 \( 1 + (0.965 - 0.262i)T \)
67 \( 1 + (-0.377 - 0.925i)T \)
71 \( 1 + (-0.818 + 0.574i)T \)
73 \( 1 + (-0.699 - 0.714i)T \)
79 \( 1 + (0.242 + 0.970i)T \)
83 \( 1 + (-0.557 - 0.830i)T \)
89 \( 1 + (0.470 - 0.882i)T \)
97 \( 1 + (0.953 - 0.301i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.39819237354091839033696347016, −18.71955366147548085566953306906, −17.97053195659289685824677196647, −17.45808473282369300283805573429, −16.44997479853572916593865368866, −15.854939445785706406003429580418, −14.92903569975020321897913590784, −14.49747565303157712685804696120, −13.45451955817729915919718053221, −12.88523449790716247917261244386, −12.500483323610037401695835507478, −11.480014799107973762583645488735, −10.761140649607404024084910990347, −10.16631353445687448453198821456, −8.96148044628588697204802720463, −8.10098562994391898742705651983, −7.79588300679475117920353641901, −7.1464898985954121898020060643, −6.28233318772125723202423814476, −5.37108030715970402571735019063, −4.26161820170377157006540384436, −3.58752009948092411661711132408, −2.86664441592592544648011220411, −1.71280907440288267192025835649, −0.85182770339828056070747084974, 0.27004734485189269200693858174, 2.088473721206108021399228771202, 2.81872235768897258058915082429, 3.480892269563722029453523656668, 4.45926556157269172167509441654, 4.992388142460211908258680228684, 5.83995439316259488589540786402, 6.92829417470548807060473757524, 7.74136630610093630856356790294, 8.70467483788767254253226197069, 8.861415819313406225902504822976, 9.81065651520467305730311283907, 10.752582445682777170673974520388, 11.51793545420240874629409582382, 11.68444421655510757777449187855, 12.99800302845989703925952532249, 13.5887475568770643957492791536, 14.54103252645878184467618357342, 15.221369375415521616693211764044, 15.71713209146900133156488975844, 16.16160076395398563065565171188, 16.8613503718462118446599822310, 18.15943074633971674226757668226, 18.57864201074438497362577145541, 19.33372975536812554354466598686

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