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Properties

Label 1-73-73.31-r1-0-0
Degree $1$
Conductor $73$
Sign $0.0242 + 0.999i$
Analytic cond. $7.84493$
Root an. cond. $7.84493$
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  + (0.173 + 0.984i)2-s + (0.866 − 0.5i)3-s + (−0.939 + 0.342i)4-s + (0.573 + 0.819i)5-s + (0.642 + 0.766i)6-s + (0.965 + 0.258i)7-s + (−0.5 − 0.866i)8-s + (0.5 − 0.866i)9-s + (−0.707 + 0.707i)10-s + (−0.819 + 0.573i)11-s + (−0.642 + 0.766i)12-s + (0.996 + 0.0871i)13-s + (−0.0871 + 0.996i)14-s + (0.906 + 0.422i)15-s + (0.766 − 0.642i)16-s + (0.258 + 0.965i)17-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)2-s + (0.866 − 0.5i)3-s + (−0.939 + 0.342i)4-s + (0.573 + 0.819i)5-s + (0.642 + 0.766i)6-s + (0.965 + 0.258i)7-s + (−0.5 − 0.866i)8-s + (0.5 − 0.866i)9-s + (−0.707 + 0.707i)10-s + (−0.819 + 0.573i)11-s + (−0.642 + 0.766i)12-s + (0.996 + 0.0871i)13-s + (−0.0871 + 0.996i)14-s + (0.906 + 0.422i)15-s + (0.766 − 0.642i)16-s + (0.258 + 0.965i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0242 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0242 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(73\)
Sign: $0.0242 + 0.999i$
Analytic conductor: \(7.84493\)
Root analytic conductor: \(7.84493\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{73} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 73,\ (1:\ ),\ 0.0242 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.772098786 + 1.729697753i\)
\(L(\frac12)\) \(\approx\) \(1.772098786 + 1.729697753i\)
\(L(1)\) \(\approx\) \(1.417697572 + 0.8332558066i\)
\(L(1)\) \(\approx\) \(1.417697572 + 0.8332558066i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad73 \( 1 \)
good2 \( 1 + (0.173 + 0.984i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (0.573 + 0.819i)T \)
7 \( 1 + (0.965 + 0.258i)T \)
11 \( 1 + (-0.819 + 0.573i)T \)
13 \( 1 + (0.996 + 0.0871i)T \)
17 \( 1 + (0.258 + 0.965i)T \)
19 \( 1 + (-0.984 + 0.173i)T \)
23 \( 1 + (0.984 + 0.173i)T \)
29 \( 1 + (-0.573 + 0.819i)T \)
31 \( 1 + (-0.422 - 0.906i)T \)
37 \( 1 + (0.173 - 0.984i)T \)
41 \( 1 + (-0.766 - 0.642i)T \)
43 \( 1 + (0.258 - 0.965i)T \)
47 \( 1 + (-0.0871 - 0.996i)T \)
53 \( 1 + (0.819 + 0.573i)T \)
59 \( 1 + (0.0871 - 0.996i)T \)
61 \( 1 + (0.642 - 0.766i)T \)
67 \( 1 + (-0.642 - 0.766i)T \)
71 \( 1 + (-0.173 - 0.984i)T \)
79 \( 1 + (0.642 + 0.766i)T \)
83 \( 1 + (-0.707 + 0.707i)T \)
89 \( 1 + (0.766 - 0.642i)T \)
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

−31.05165087306135102578381702332, −30.06584252344147484775870214929, −28.874632470324320021773068010953, −27.76336964564911170157826742753, −27.01125137095287228134840448334, −25.72080901635319903742532087086, −24.44346800103705888358539558642, −23.30357822270577273842979182338, −21.601373415454512089274986876019, −20.89368635914752865160532154014, −20.485544315986140196835943026526, −19.033805419800489871134312681497, −17.91018863056632305095466255167, −16.44298516932156011927259187094, −14.885253448587263716198659192848, −13.670882449344360022011041423647, −13.10199412788772635866675811185, −11.28079169864561737529782145306, −10.25113795488910489375980821443, −8.93456050976894989414411519686, −8.18813484715429594902121210046, −5.3413508892361201757590218635, −4.36077520970511244473847430318, −2.75402789995466680765011341535, −1.286565047190803385835544316090, 1.98186430571076637991483753077, 3.74488322870825446625244458778, 5.581524186698114960248078171113, 6.90687451449024415045886206644, 7.98603109565164868736430793183, 9.03147938858582357227968299030, 10.63630019478262724346305539936, 12.71085718080869383652907850369, 13.66032633473841716755372348360, 14.788820500437367132006951198859, 15.24765277868233713999953124913, 17.21039964043699396960730458515, 18.24685786615340291902117863278, 18.82170795024235852992628750353, 20.82779827100279448980097872126, 21.59183494827164065634934356315, 23.223264864488291862680504618047, 23.97880794817102475989281310180, 25.33146537988320602762867485133, 25.76295942686072448567145607569, 26.75892828986784336548649635744, 28.064126009691171201262585547, 29.85961710678250401481740915688, 30.79521511863272565101072908684, 31.35331861454611749099102121725

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