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Properties

Label 1-73-73.3-r0-0-0
Degree $1$
Conductor $73$
Sign $0.221 - 0.975i$
Analytic cond. $0.339010$
Root an. cond. $0.339010$
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  + (−0.5 − 0.866i)2-s − 3-s + (−0.5 + 0.866i)4-s + (0.866 + 0.5i)5-s + (0.5 + 0.866i)6-s − i·7-s + 8-s + 9-s − i·10-s + (−0.866 − 0.5i)11-s + (0.5 − 0.866i)12-s + (0.866 − 0.5i)13-s + (−0.866 + 0.5i)14-s + (−0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s − i·17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s − 3-s + (−0.5 + 0.866i)4-s + (0.866 + 0.5i)5-s + (0.5 + 0.866i)6-s − i·7-s + 8-s + 9-s − i·10-s + (−0.866 − 0.5i)11-s + (0.5 − 0.866i)12-s + (0.866 − 0.5i)13-s + (−0.866 + 0.5i)14-s + (−0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s − i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.221 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.221 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(73\)
Sign: $0.221 - 0.975i$
Analytic conductor: \(0.339010\)
Root analytic conductor: \(0.339010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{73} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 73,\ (0:\ ),\ 0.221 - 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4721851865 - 0.3770565214i\)
\(L(\frac12)\) \(\approx\) \(0.4721851865 - 0.3770565214i\)
\(L(1)\) \(\approx\) \(0.6209807608 - 0.2955215396i\)
\(L(1)\) \(\approx\) \(0.6209807608 - 0.2955215396i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad73 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 - T \)
5 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 - iT \)
11 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (0.866 - 0.5i)T \)
17 \( 1 - iT \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.866 - 0.5i)T \)
31 \( 1 + (0.866 - 0.5i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + iT \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 + (-0.866 + 0.5i)T \)
59 \( 1 + (-0.866 + 0.5i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 - iT \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 - T \)
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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

−32.22493892912364891040123113437, −30.79679632653301897222913284425, −28.91518517339712775619455983468, −28.53610400944618818034967651865, −27.733254614310601754606649805646, −26.16545903246175327333446857231, −25.2781536411533185183372001834, −24.17490554272287161698665643366, −23.3656186177588949039923284898, −22.01792142322986065832252836393, −21.07783899992964043088284331596, −19.10693087503411681182305116605, −17.9572198257893929913281181847, −17.47580683152238679568175043249, −16.10916446220135113740207847907, −15.4114420104460124464396099209, −13.64137862216255632412376893013, −12.52020832009627036210620314317, −10.84385341947634911165803031616, −9.66165235718157806481538816421, −8.53387734573114366154054599251, −6.773944509250538561773653524479, −5.69028580938628171766986622197, −4.905719286990377214869652426434, −1.624834420935877239359319557, 1.105548103503191266759166395004, 3.0882748826154075243761280197, 4.86086835615036005842402999751, 6.45919541417415273165456217389, 7.922893879948529428874916560593, 9.89467056120818194325882763969, 10.51995645313035945687888090623, 11.47017599002202742908297317971, 13.04553366123105862452657844254, 13.798872752966805684584501494482, 16.09814199126262041967999724017, 17.08302554035289982451558299807, 18.106366362053586945401707755136, 18.72562770635795206209280366846, 20.582099700280692094146562729232, 21.227071325681926943080475892367, 22.5835827788173017063128680167, 23.15308702705209509389523605080, 24.92644810776996185969307934941, 26.38413949764254478374419547987, 27.020315917473613738797270744082, 28.359364372433243133050826946094, 29.2901066313206505172429934235, 29.755356359791275057972039688819, 30.85078751510172095077463505912

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