Properties

Label 1-73-73.25-r0-0-0
Degree $1$
Conductor $73$
Sign $-0.341 + 0.939i$
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  + (0.173 + 0.984i)2-s + (0.5 + 0.866i)3-s + (−0.939 + 0.342i)4-s + (0.984 + 0.173i)5-s + (−0.766 + 0.642i)6-s + (0.866 − 0.5i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)9-s + i·10-s + (−0.984 − 0.173i)11-s + (−0.766 − 0.642i)12-s + (−0.642 − 0.766i)13-s + (0.642 + 0.766i)14-s + (0.342 + 0.939i)15-s + (0.766 − 0.642i)16-s + (−0.866 − 0.5i)17-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)2-s + (0.5 + 0.866i)3-s + (−0.939 + 0.342i)4-s + (0.984 + 0.173i)5-s + (−0.766 + 0.642i)6-s + (0.866 − 0.5i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)9-s + i·10-s + (−0.984 − 0.173i)11-s + (−0.766 − 0.642i)12-s + (−0.642 − 0.766i)13-s + (0.642 + 0.766i)14-s + (0.342 + 0.939i)15-s + (0.766 − 0.642i)16-s + (−0.866 − 0.5i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.341 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.341 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(73\)
Sign: $-0.341 + 0.939i$
Analytic conductor: \(0.339010\)
Root analytic conductor: \(0.339010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{73} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 73,\ (0:\ ),\ -0.341 + 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6913736680 + 0.9868158741i\)
\(L(\frac12)\) \(\approx\) \(0.6913736680 + 0.9868158741i\)
\(L(1)\) \(\approx\) \(0.9623972130 + 0.8287528548i\)
\(L(1)\) \(\approx\) \(0.9623972130 + 0.8287528548i\)

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 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (0.984 + 0.173i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (-0.984 - 0.173i)T \)
13 \( 1 + (-0.642 - 0.766i)T \)
17 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (-0.173 - 0.984i)T \)
23 \( 1 + (-0.173 + 0.984i)T \)
29 \( 1 + (0.984 - 0.173i)T \)
31 \( 1 + (-0.342 + 0.939i)T \)
37 \( 1 + (0.173 - 0.984i)T \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 + (0.642 - 0.766i)T \)
53 \( 1 + (-0.984 + 0.173i)T \)
59 \( 1 + (0.642 + 0.766i)T \)
61 \( 1 + (-0.766 - 0.642i)T \)
67 \( 1 + (-0.766 + 0.642i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
79 \( 1 + (-0.766 + 0.642i)T \)
83 \( 1 + iT \)
89 \( 1 + (0.766 - 0.642i)T \)
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

−31.12001591063411454613534076888, −30.173590608150789307976045531410, −29.0140173159273304410634930606, −28.59973930855340135143154642397, −26.93472875118271485633770873781, −25.81349043751523923881711821951, −24.48063717434876092670179903982, −23.75278341080433804785208293646, −22.14907048650128755211376525893, −21.071997902503011196681480034066, −20.39889911458409080792497491926, −18.92376341555083479986857790569, −18.18214397813383616860151374781, −17.272124703708146130511346990274, −14.82054404453732169302281288223, −13.981304655201834828868861487611, −12.914963229076726738433226957285, −12.01220073399764107135714310615, −10.5061090896768132678679286995, −9.14409246258856787979071021611, −8.11647122259941668553084986949, −6.10797863712511526899942491721, −4.69198806217840398388079704231, −2.50366557400733554131221051936, −1.78694550477054731236648099799, 2.77413689151156307998761059662, 4.65683355387168966916747056196, 5.450005811006510030964209790625, 7.29353206867959404031564653356, 8.50833201384972350589236118656, 9.72733804362528380960361035385, 10.82768831954156710257134961985, 13.206236152621971907791761472231, 13.98402905681626435690022794480, 15.00678095568932566217583434688, 15.9971272958999606550489643825, 17.384057420945989551941898917940, 17.98221642224176492534975935790, 19.929281181828411212083976132970, 21.29572371381663108893269994748, 21.84329186645167318369101324746, 23.20104147796738724904824260546, 24.52610487350577888620894558584, 25.3996294914013790761278097416, 26.47977477968823457329641457716, 27.04548760853979249667907051765, 28.36507040026811244041703619328, 29.98280392155666871966183704303, 31.1551443925697697526759472057, 32.13152810918228300486979487736

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