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Properties

Label 1-73-73.71-r0-0-0
Degree $1$
Conductor $73$
Sign $-0.625 - 0.780i$
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  + (0.766 − 0.642i)2-s + (−0.5 − 0.866i)3-s + (0.173 − 0.984i)4-s + (−0.766 − 0.642i)5-s + (−0.939 − 0.342i)6-s + (0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)9-s − 10-s + (−0.766 − 0.642i)11-s + (−0.939 + 0.342i)12-s + (0.939 + 0.342i)13-s + (0.939 + 0.342i)14-s + (−0.173 + 0.984i)15-s + (−0.939 − 0.342i)16-s + (0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (−0.5 − 0.866i)3-s + (0.173 − 0.984i)4-s + (−0.766 − 0.642i)5-s + (−0.939 − 0.342i)6-s + (0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)9-s − 10-s + (−0.766 − 0.642i)11-s + (−0.939 + 0.342i)12-s + (0.939 + 0.342i)13-s + (0.939 + 0.342i)14-s + (−0.173 + 0.984i)15-s + (−0.939 − 0.342i)16-s + (0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.625 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.625 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(73\)
Sign: $-0.625 - 0.780i$
Analytic conductor: \(0.339010\)
Root analytic conductor: \(0.339010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{73} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 73,\ (0:\ ),\ -0.625 - 0.780i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4645931974 - 0.9672476183i\)
\(L(\frac12)\) \(\approx\) \(0.4645931974 - 0.9672476183i\)
\(L(1)\) \(\approx\) \(0.8512802287 - 0.7978279313i\)
\(L(1)\) \(\approx\) \(0.8512802287 - 0.7978279313i\)

Euler product

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

−32.12618511890937800037109731843, −30.87834669209433539402897828325, −30.23612784940538771009679492564, −28.65988425663432883944874064296, −27.24719266396296875516595884596, −26.55744576436288151186622789199, −25.60527167435883585368512247130, −23.76673491260658911920947668815, −23.20548747945078816100308375316, −22.44899103879053740904560084116, −21.033078692731196654115821304191, −20.36104432067221994421774965177, −18.26434617975847624110294435734, −17.105919153297162656394912231138, −16.004963539552691236884995338142, −15.14296423070222564079294160652, −14.211657475356471862671587128215, −12.59522629537839355444545836981, −11.25938914884399467915868489981, −10.403888925125762407928570395173, −8.24928838434111408590445828810, −7.126820282109225422550562090077, −5.63172151291345604504546700720, −4.297217919811487437211000281309, −3.38099826304543824827007064154, 1.211614497409214702136771987808, 2.974168433559201374524867461961, 4.92752509851659583223633327712, 5.789557094627061876237838912706, 7.55881202508332896333668015779, 8.99960023558562826633874907839, 11.2363352085244977543057643915, 11.605974609418195149760546961251, 12.84493400854463604031389821327, 13.71500817627322114797787006924, 15.34221867647587788615464562902, 16.396569648860598266436105548658, 18.347798253046601736912407755557, 18.86071163624187291273126988996, 20.220264042489960534443398561, 21.258703599371862132151200878003, 22.52770898354656565147990984161, 23.67110818559340328164351537681, 24.13151154712835290132577204454, 25.23419384242921476534086907775, 27.38627319390518292025712588171, 28.31026382744422772436643612105, 28.970633734358297757071310706972, 30.22165801128715767594442468319, 31.2728147591660400613844860535

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