Properties

Degree 1
Conductor 73
Sign $-0.625 - 0.780i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(73\)
\( \varepsilon \)  =  $-0.625 - 0.780i$
motivic weight  =  \(0\)
character  :  $\chi_{73} (71, \cdot )$
Sato-Tate  :  $\mu(18)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 73,\ (0:\ ),\ -0.625 - 0.780i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.4645931974 - 0.9672476183i$
$L(\frac12,\chi)$  $\approx$  $0.4645931974 - 0.9672476183i$
$L(\chi,1)$  $\approx$  0.8512802287 - 0.7978279313i
$L(1,\chi)$  $\approx$  0.8512802287 - 0.7978279313i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

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