Properties

Label 1-73-73.39-r1-0-0
Degree $1$
Conductor $73$
Sign $0.997 + 0.0652i$
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 + (0.819 − 0.573i)5-s + (−0.642 − 0.766i)6-s + (0.258 − 0.965i)7-s + (−0.5 − 0.866i)8-s + (0.5 − 0.866i)9-s + (0.707 + 0.707i)10-s + (−0.573 − 0.819i)11-s + (0.642 − 0.766i)12-s + (−0.0871 + 0.996i)13-s + (0.996 + 0.0871i)14-s + (−0.422 + 0.906i)15-s + (0.766 − 0.642i)16-s + (0.965 − 0.258i)17-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)2-s + (−0.866 + 0.5i)3-s + (−0.939 + 0.342i)4-s + (0.819 − 0.573i)5-s + (−0.642 − 0.766i)6-s + (0.258 − 0.965i)7-s + (−0.5 − 0.866i)8-s + (0.5 − 0.866i)9-s + (0.707 + 0.707i)10-s + (−0.573 − 0.819i)11-s + (0.642 − 0.766i)12-s + (−0.0871 + 0.996i)13-s + (0.996 + 0.0871i)14-s + (−0.422 + 0.906i)15-s + (0.766 − 0.642i)16-s + (0.965 − 0.258i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0652i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0652i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(73\)
Sign: $0.997 + 0.0652i$
Analytic conductor: \(7.84493\)
Root analytic conductor: \(7.84493\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{73} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 73,\ (1:\ ),\ 0.997 + 0.0652i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.228272370 + 0.04009230928i\)
\(L(\frac12)\) \(\approx\) \(1.228272370 + 0.04009230928i\)
\(L(1)\) \(\approx\) \(0.9092458760 + 0.2608289407i\)
\(L(1)\) \(\approx\) \(0.9092458760 + 0.2608289407i\)

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 \( 1 + (0.819 - 0.573i)T \)
7 \( 1 + (0.258 - 0.965i)T \)
11 \( 1 + (-0.573 - 0.819i)T \)
13 \( 1 + (-0.0871 + 0.996i)T \)
17 \( 1 + (0.965 - 0.258i)T \)
19 \( 1 + (0.984 - 0.173i)T \)
23 \( 1 + (-0.984 - 0.173i)T \)
29 \( 1 + (-0.819 - 0.573i)T \)
31 \( 1 + (0.906 - 0.422i)T \)
37 \( 1 + (0.173 - 0.984i)T \)
41 \( 1 + (-0.766 - 0.642i)T \)
43 \( 1 + (0.965 + 0.258i)T \)
47 \( 1 + (0.996 - 0.0871i)T \)
53 \( 1 + (0.573 - 0.819i)T \)
59 \( 1 + (-0.996 - 0.0871i)T \)
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

−30.80528440539298905219210610978, −30.13489890309399417921502861269, −29.12569549592030923983931286598, −28.312956058776125359977341563863, −27.48873835708692952171549439205, −25.818148901507773952496188933115, −24.62138361602023875293269418952, −23.23186152409539811093474055065, −22.33755511736744594908772961729, −21.61344766015107582784829992654, −20.37009662912478619718663677862, −18.67886731558557073719923885587, −18.20240377662440131329406079057, −17.31751300749365924544772957435, −15.31220182354905810643618357196, −13.94151023039828381301149109388, −12.709864244676229927709775738272, −11.92584374158121047016139427496, −10.57246745577529621380146665928, −9.75232537589226257480050248522, −7.81875961186345441915996901860, −5.87050762631459034305759025157, −5.149506708148832745311680191098, −2.81179015656353831357605455517, −1.53830056324085977662934489009, 0.7098013418337631384032358719, 3.99186288387699754733042743841, 5.19103013013767147390746593697, 6.128014774535129492593825676818, 7.59195529789751413007912439626, 9.25126014282636731382569382841, 10.26869200360383660433843663292, 11.93340507465931705140260056286, 13.43662245961357318907025821559, 14.24034846227737106087247289034, 16.01255904375447567698584224456, 16.63059107249090055350668006424, 17.46356982787354727169741797981, 18.5666454634284079326105906833, 20.77040051658365337175513533162, 21.54660319311385163237245558526, 22.69187059074762737577574688759, 23.86531228315901658715047854676, 24.38463687447556359888680862241, 26.08333731752029041580895542846, 26.72150595270676526152225756255, 27.95183004859458276576213289246, 29.01735165872380860945549246582, 30.11903733782889640884421772474, 31.88325645887990768903539555710

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