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Properties

Label 1-73-73.57-r0-0-0
Degree $1$
Conductor $73$
Sign $0.372 - 0.928i$
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  + (−0.939 − 0.342i)2-s + (−0.5 − 0.866i)3-s + (0.766 + 0.642i)4-s + (0.939 − 0.342i)5-s + (0.173 + 0.984i)6-s + (0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)9-s − 10-s + (0.939 − 0.342i)11-s + (0.173 − 0.984i)12-s + (−0.173 − 0.984i)13-s + (−0.173 − 0.984i)14-s + (−0.766 − 0.642i)15-s + (0.173 + 0.984i)16-s + (0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)2-s + (−0.5 − 0.866i)3-s + (0.766 + 0.642i)4-s + (0.939 − 0.342i)5-s + (0.173 + 0.984i)6-s + (0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)9-s − 10-s + (0.939 − 0.342i)11-s + (0.173 − 0.984i)12-s + (−0.173 − 0.984i)13-s + (−0.173 − 0.984i)14-s + (−0.766 − 0.642i)15-s + (0.173 + 0.984i)16-s + (0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.372 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.372 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(73\)
Sign: $0.372 - 0.928i$
Analytic conductor: \(0.339010\)
Root analytic conductor: \(0.339010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{73} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 73,\ (0:\ ),\ 0.372 - 0.928i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5447194924 - 0.3683897425i\)
\(L(\frac12)\) \(\approx\) \(0.5447194924 - 0.3683897425i\)
\(L(1)\) \(\approx\) \(0.6707072878 - 0.2856322957i\)
\(L(1)\) \(\approx\) \(0.6707072878 - 0.2856322957i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad73 \( 1 \)
good2 \( 1 + (-0.939 - 0.342i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (0.939 - 0.342i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (0.939 - 0.342i)T \)
13 \( 1 + (-0.173 - 0.984i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.939 - 0.342i)T \)
23 \( 1 + (-0.939 + 0.342i)T \)
29 \( 1 + (0.939 + 0.342i)T \)
31 \( 1 + (-0.766 + 0.642i)T \)
37 \( 1 + (-0.939 + 0.342i)T \)
41 \( 1 + (0.173 - 0.984i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (-0.173 + 0.984i)T \)
53 \( 1 + (0.939 + 0.342i)T \)
59 \( 1 + (-0.173 - 0.984i)T \)
61 \( 1 + (0.173 - 0.984i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (-0.939 - 0.342i)T \)
79 \( 1 + (0.173 + 0.984i)T \)
83 \( 1 - T \)
89 \( 1 + (0.173 + 0.984i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−32.366827311950687074646561230875, −30.224751865674837817177656469348, −29.37924214088887883596394346396, −28.28019203581044351885894096771, −27.40286251113687697793572086971, −26.372720424353123817240599744695, −25.68380259997038255209436073693, −24.23678944302345618428344364606, −23.11627331859789696848709753637, −21.66138551895614513410179941128, −20.800145660594765169796314216605, −19.54130592259412820374185438742, −18.01932860257698865302921976292, −17.06334260048832662126982655039, −16.62786360063819328563728321733, −14.867072957032162325288348938530, −14.20881809897588718114088113280, −11.84858146420048775092823402944, −10.58252188404683406997358854211, −9.93794561731937766333027188921, −8.72340453197897283811485918540, −6.888040397078551896481769201433, −5.89658941840995539063763612342, −4.18672555913364489328130565649, −1.75063878534414692448475476561, 1.32260472351175958574002864752, 2.57189029717386516051677176692, 5.442931052225130753997488468345, 6.586215147224556327598185118776, 8.11945953515622108578471065966, 9.156680003313019725167494992456, 10.62186389846994701946356353994, 11.90115450744842482376384433059, 12.69399189505275428239284479086, 14.22540039190242202814903375904, 16.077017876689923705611267942293, 17.3700683912114186338773689000, 17.835159913334970637144047343990, 18.91977172692724816125764788625, 20.07118101273198550499325688374, 21.41145818591562190397582523727, 22.30394137527018385687695913693, 24.19338673942548449362194993314, 25.04000884065610436400168679631, 25.55876359215479125721716786719, 27.56716481419901023676201183489, 27.995382285497593810259970899050, 29.35344719538747063507664391459, 29.76966793301100250330465222654, 30.91380103986796167570166051978

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