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Properties

Label 1-6040-6040.1789-r0-0-0
Degree $1$
Conductor $6040$
Sign $-0.992 - 0.125i$
Analytic cond. $28.0496$
Root an. cond. $28.0496$
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  + (−0.809 − 0.587i)3-s + (−0.669 + 0.743i)7-s + (0.309 + 0.951i)9-s + (−0.913 − 0.406i)11-s + (0.913 − 0.406i)13-s + (0.978 − 0.207i)17-s − 19-s + (0.978 − 0.207i)21-s + (0.5 + 0.866i)23-s + (0.309 − 0.951i)27-s + (0.809 − 0.587i)29-s + (−0.978 + 0.207i)31-s + (0.5 + 0.866i)33-s + (−0.104 + 0.994i)37-s + (−0.978 − 0.207i)39-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)3-s + (−0.669 + 0.743i)7-s + (0.309 + 0.951i)9-s + (−0.913 − 0.406i)11-s + (0.913 − 0.406i)13-s + (0.978 − 0.207i)17-s − 19-s + (0.978 − 0.207i)21-s + (0.5 + 0.866i)23-s + (0.309 − 0.951i)27-s + (0.809 − 0.587i)29-s + (−0.978 + 0.207i)31-s + (0.5 + 0.866i)33-s + (−0.104 + 0.994i)37-s + (−0.978 − 0.207i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6040\)    =    \(2^{3} \cdot 5 \cdot 151\)
Sign: $-0.992 - 0.125i$
Analytic conductor: \(28.0496\)
Root analytic conductor: \(28.0496\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6040} (1789, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6040,\ (0:\ ),\ -0.992 - 0.125i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.007018143606 - 0.1117139375i\)
\(L(\frac12)\) \(\approx\) \(0.007018143606 - 0.1117139375i\)
\(L(1)\) \(\approx\) \(0.6565442033 - 0.06180225340i\)
\(L(1)\) \(\approx\) \(0.6565442033 - 0.06180225340i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
151 \( 1 \)
good3 \( 1 + (-0.809 - 0.587i)T \)
7 \( 1 + (-0.669 + 0.743i)T \)
11 \( 1 + (-0.913 - 0.406i)T \)
13 \( 1 + (0.913 - 0.406i)T \)
17 \( 1 + (0.978 - 0.207i)T \)
19 \( 1 - T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (-0.978 + 0.207i)T \)
37 \( 1 + (-0.104 + 0.994i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (0.669 + 0.743i)T \)
47 \( 1 + (-0.913 + 0.406i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 - T \)
61 \( 1 + (0.104 + 0.994i)T \)
67 \( 1 + (0.309 - 0.951i)T \)
71 \( 1 + (-0.978 - 0.207i)T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (0.309 - 0.951i)T \)
89 \( 1 + (0.669 + 0.743i)T \)
97 \( 1 + (0.978 - 0.207i)T \)
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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

−17.91936780709830916311025055629, −17.235203045443783120242000046051, −16.61909798918647100169834921559, −16.14165354105693204885885457543, −15.66557367462902719859224506012, −14.746770255580019584117324643313, −14.260066624962036493629104634, −13.129707628064223290005058921562, −12.83255959112797385792984354110, −12.15467478761351342920326141844, −11.22173979557148073554451001262, −10.53899342603206528670504485480, −10.38155658776288585254189323574, −9.52003959785588352124377029916, −8.80768297310659240265209849648, −7.96788125894231992267044629200, −7.051666761464624568043744045399, −6.53333328246316330449415325688, −5.85826773246119082233316254675, −5.06761676700063307403930007270, −4.39679604691707967906552776632, −3.68037164261810597364279789344, −3.08824671283165726038867200591, −1.892065423057326724246498700149, −0.90607201346506880640039480683, 0.04047136207293107865157928707, 1.10254893947228872078553769306, 1.92391375820860014461133057885, 2.91973049919139936935378847795, 3.39229668023818804447611444361, 4.65495206421245723366412566928, 5.31558474334423508844858253523, 6.02070717961062952363731610389, 6.29217408750546041506483347795, 7.30280538597003340288689720688, 7.97174940175892120432780792951, 8.568323820162852600381911856508, 9.44675166978043139273671375118, 10.29882261351240319452477943886, 10.78071916914403522184296235057, 11.50367430080721477366617181448, 12.19941547049957623972414718726, 12.77649155608741130972056086551, 13.30146779591741232645892677568, 13.841408568253724160504739833868, 14.9429965374361722947017089037, 15.61200584825208646882553121251, 16.10922697675230126162520647873, 16.70258609857644273881848419898, 17.45146181336534008172904458279

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