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Properties

Label 1-73-73.46-r0-0-0
Degree $1$
Conductor $73$
Sign $0.821 - 0.569i$
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  + 2-s − 3-s + 4-s − i·5-s − 6-s − i·7-s + 8-s + 9-s − i·10-s + i·11-s − 12-s + i·13-s − i·14-s + i·15-s + 16-s − i·17-s + ⋯
L(s)  = 1  + 2-s − 3-s + 4-s − i·5-s − 6-s − i·7-s + 8-s + 9-s − i·10-s + i·11-s − 12-s + i·13-s − i·14-s + i·15-s + 16-s − i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.821 - 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.821 - 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(73\)
Sign: $0.821 - 0.569i$
Analytic conductor: \(0.339010\)
Root analytic conductor: \(0.339010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{73} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 73,\ (0:\ ),\ 0.821 - 0.569i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.219098157 - 0.3811308268i\)
\(L(\frac12)\) \(\approx\) \(1.219098157 - 0.3811308268i\)
\(L(1)\) \(\approx\) \(1.329111003 - 0.2481139176i\)
\(L(1)\) \(\approx\) \(1.329111003 - 0.2481139176i\)

Euler product

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

−31.75968538270467463017602768855, −30.28411870227525271032764356105, −29.84471633819063247576653255057, −28.640527008106417334960835904361, −27.59134057115339599906494652404, −26.063917297517976118692035208411, −24.82006676713214580977018570606, −23.79706317124314918744500544755, −22.733881349086565710057876866997, −21.93132805035268198657207997316, −21.33501643103057886520607529782, −19.38546574780285458737739463289, −18.35980419775882984032326531617, −16.99106267379733547748173522672, −15.606923735835097755345593976525, −14.93173721561836354755184930180, −13.35530465310146986552723117882, −12.20365216705243329810464080925, −11.17736526858127097635300221417, −10.31167921046802619995626573471, −7.87079017798354200088711094583, −6.12263902746872381620441166583, −5.85164043939938542528754665901, −3.96284879488798697692188724283, −2.38946164260758975988960520196, 1.59350807350854202222056631193, 4.25479917828765405453596586436, 4.8141461369410578590685447179, 6.39336796228355961047496667240, 7.484362790349082798245899059939, 9.75840046932713984925978796655, 11.08725756650962312830614004212, 12.21230614299944400217041420567, 13.025156886890388894926797084762, 14.30409862965074292200999716529, 16.00939146873952251325642889881, 16.58824431288525457127562047944, 17.70793449152811477083947379383, 19.67166009981829259250219767388, 20.681771994811057999366984529984, 21.63364113432615535509673956894, 22.981900761738847378999094777253, 23.58793836628749446994122219436, 24.43691179620743580352379016446, 25.76269975130847993869960132048, 27.43644371023506397099192644768, 28.53899339147749892342996127338, 29.30970591544731562956577753021, 30.299821900438690613644963085287, 31.55632347102447731882666358546

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