Properties

Label 1-2368-2368.611-r0-0-0
Degree $1$
Conductor $2368$
Sign $0.875 - 0.484i$
Analytic cond. $10.9969$
Root an. cond. $10.9969$
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  + (0.537 − 0.843i)3-s + (0.953 + 0.300i)5-s + (−0.996 + 0.0871i)7-s + (−0.422 − 0.906i)9-s + (0.793 + 0.608i)11-s + (0.999 + 0.0436i)13-s + (0.766 − 0.642i)15-s + (0.939 + 0.342i)17-s + (−0.537 + 0.843i)19-s + (−0.461 + 0.887i)21-s + (−0.258 − 0.965i)23-s + (0.819 + 0.573i)25-s + (−0.991 − 0.130i)27-s + (0.991 − 0.130i)29-s − i·31-s + ⋯
L(s)  = 1  + (0.537 − 0.843i)3-s + (0.953 + 0.300i)5-s + (−0.996 + 0.0871i)7-s + (−0.422 − 0.906i)9-s + (0.793 + 0.608i)11-s + (0.999 + 0.0436i)13-s + (0.766 − 0.642i)15-s + (0.939 + 0.342i)17-s + (−0.537 + 0.843i)19-s + (−0.461 + 0.887i)21-s + (−0.258 − 0.965i)23-s + (0.819 + 0.573i)25-s + (−0.991 − 0.130i)27-s + (0.991 − 0.130i)29-s − i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.875 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.875 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2368\)    =    \(2^{6} \cdot 37\)
Sign: $0.875 - 0.484i$
Analytic conductor: \(10.9969\)
Root analytic conductor: \(10.9969\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2368} (611, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2368,\ (0:\ ),\ 0.875 - 0.484i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.395433700 - 0.6183799369i\)
\(L(\frac12)\) \(\approx\) \(2.395433700 - 0.6183799369i\)
\(L(1)\) \(\approx\) \(1.475029481 - 0.2792925687i\)
\(L(1)\) \(\approx\) \(1.475029481 - 0.2792925687i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
37 \( 1 \)
good3 \( 1 + (0.537 - 0.843i)T \)
5 \( 1 + (0.953 + 0.300i)T \)
7 \( 1 + (-0.996 + 0.0871i)T \)
11 \( 1 + (0.793 + 0.608i)T \)
13 \( 1 + (0.999 + 0.0436i)T \)
17 \( 1 + (0.939 + 0.342i)T \)
19 \( 1 + (-0.537 + 0.843i)T \)
23 \( 1 + (-0.258 - 0.965i)T \)
29 \( 1 + (0.991 - 0.130i)T \)
31 \( 1 - iT \)
41 \( 1 + (-0.906 - 0.422i)T \)
43 \( 1 + (-0.382 - 0.923i)T \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + (-0.461 + 0.887i)T \)
59 \( 1 + (-0.461 + 0.887i)T \)
61 \( 1 + (-0.0436 + 0.999i)T \)
67 \( 1 + (0.461 + 0.887i)T \)
71 \( 1 + (0.573 + 0.819i)T \)
73 \( 1 + (-0.707 - 0.707i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (0.675 - 0.737i)T \)
89 \( 1 + (0.996 + 0.0871i)T \)
97 \( 1 + (0.866 + 0.5i)T \)
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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

−19.76379635876273166443768422417, −19.08256247559432784585008765665, −18.24102213063979090469434768091, −17.237025032699603643435933952675, −16.718716652168984304772696534912, −16.014828941565138527425414064304, −15.54469616164082119482027132869, −14.42056360460686465374662726820, −13.86648082304553205718994040563, −13.39715790082026105017688469539, −12.5632396101196174696856214992, −11.51633676280528069431667270402, −10.6916983123881644429851788536, −9.96512072121315780617417636684, −9.41329505408497527923888034422, −8.805358527066810389214527356691, −8.14407666660086668095022077798, −6.79223614293331731413887736161, −6.197360429726033176537960939010, −5.3851666472588275905342515335, −4.58166433963280934636735007000, −3.34633750121555464412260029264, −3.26501905386065745351668090168, −1.978095184275889356762737259610, −0.94421333525309802510709993411, 0.97615195429370766830881703885, 1.792139477683350469829665411230, 2.583900255017008596657362769987, 3.43404810560812050432801495234, 4.17607483311816051863731749557, 5.76529367896032377330985418121, 6.19111175962871891093303309935, 6.73018722121320243784147615457, 7.60378706174728325740699246251, 8.64069501827560970540514389048, 9.10864311435768236405020880656, 10.06414677902476418108083830762, 10.46161149116610839830867559111, 11.88883922822097331339458584331, 12.35180588999776502860436280465, 13.119146232115446144497913776347, 13.693281316486599581309115695792, 14.40549038901841016508130104743, 14.93954062358776273851383886080, 15.945571810245318564572437176574, 16.94846040265147493893747532717, 17.28464387758917270842365910427, 18.41342526268732103302769299759, 18.69650214835725995598002029812, 19.3194258140183283358301463366

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