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Properties

Label 1-73-73.43-r1-0-0
Degree $1$
Conductor $73$
Sign $-0.0725 - 0.997i$
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  + (−0.5 − 0.866i)2-s + i·3-s + (−0.5 + 0.866i)4-s + (−0.258 − 0.965i)5-s + (0.866 − 0.5i)6-s + (−0.707 + 0.707i)7-s + 8-s − 9-s + (−0.707 + 0.707i)10-s + (0.965 − 0.258i)11-s + (−0.866 − 0.5i)12-s + (0.258 − 0.965i)13-s + (0.965 + 0.258i)14-s + (0.965 − 0.258i)15-s + (−0.5 − 0.866i)16-s + (0.707 − 0.707i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + i·3-s + (−0.5 + 0.866i)4-s + (−0.258 − 0.965i)5-s + (0.866 − 0.5i)6-s + (−0.707 + 0.707i)7-s + 8-s − 9-s + (−0.707 + 0.707i)10-s + (0.965 − 0.258i)11-s + (−0.866 − 0.5i)12-s + (0.258 − 0.965i)13-s + (0.965 + 0.258i)14-s + (0.965 − 0.258i)15-s + (−0.5 − 0.866i)16-s + (0.707 − 0.707i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0725 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0725 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(73\)
Sign: $-0.0725 - 0.997i$
Analytic conductor: \(7.84493\)
Root analytic conductor: \(7.84493\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{73} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 73,\ (1:\ ),\ -0.0725 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5951454223 - 0.6400340568i\)
\(L(\frac12)\) \(\approx\) \(0.5951454223 - 0.6400340568i\)
\(L(1)\) \(\approx\) \(0.6946436763 - 0.2413926346i\)
\(L(1)\) \(\approx\) \(0.6946436763 - 0.2413926346i\)

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 \( 1 + iT \)
5 \( 1 + (-0.258 - 0.965i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
11 \( 1 + (0.965 - 0.258i)T \)
13 \( 1 + (0.258 - 0.965i)T \)
17 \( 1 + (0.707 - 0.707i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 + (-0.866 - 0.5i)T \)
29 \( 1 + (0.258 - 0.965i)T \)
31 \( 1 + (0.258 - 0.965i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (0.707 + 0.707i)T \)
47 \( 1 + (0.965 - 0.258i)T \)
53 \( 1 + (-0.965 - 0.258i)T \)
59 \( 1 + (-0.965 - 0.258i)T \)
61 \( 1 + (0.866 + 0.5i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (0.866 - 0.5i)T \)
83 \( 1 + (-0.707 + 0.707i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + iT \)
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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

−31.556153854043502889563327283191, −30.4126667305998712890561522781, −29.453906615008878274246145152962, −28.298787452489107052315107130672, −26.90621300014630236611115162783, −25.97476095888200994735819660872, −25.26724960282003966175288550033, −23.875311208896510901070997948710, −23.1817580139854060877657016338, −22.248750190929382317488077929218, −19.81030364142213217883921233804, −19.21460390250892316731353728636, −18.20855333094948920838482877708, −17.1437184612049879382086998039, −16.04087413780790175391312218998, −14.32548344784688983462911954299, −13.97091945219552627832952243017, −12.1901816238209347641590094000, −10.69263122252571494887512867071, −9.33094577989418921118020339743, −7.73739779197000281269188813385, −6.88161583641615574963112390467, −6.07760303813595870908445102080, −3.7012902199627287062813462322, −1.40356616528668972086398070240, 0.57695259360942913705030702833, 2.93399899406063810380022868650, 4.14994864306472026495196839378, 5.60525191375040542425583114864, 8.13555395010620054265095841497, 9.197710675361497443029846848085, 9.89984123757049218448942378727, 11.519320897231261127652260720268, 12.31316692732518182835389994048, 13.76935515769214090789396632947, 15.6271778258369551878911841245, 16.446154842697749685056635920024, 17.52409784993806805369849550680, 19.09395405863067351872104091562, 20.14056380457847091483769221600, 20.82741330368864669595064760088, 22.12325270357784262613366060057, 22.729296760950165582354883335576, 24.79485944451479462030825134290, 25.802860034736706098613280672262, 27.0816460445000695106010169550, 27.9174706388802541587336608704, 28.46273008370428481127821536116, 29.68581897157552348381296791434, 31.1216357805412912089388817421

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