Properties

Degree 1
Conductor $ 3 \cdot 29 $
Sign $0.653 - 0.757i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.974 − 0.222i)2-s + (0.900 − 0.433i)4-s + (−0.222 − 0.974i)5-s + (−0.900 − 0.433i)7-s + (0.781 − 0.623i)8-s + (−0.433 − 0.900i)10-s + (0.781 + 0.623i)11-s + (−0.623 + 0.781i)13-s + (−0.974 − 0.222i)14-s + (0.623 − 0.781i)16-s + i·17-s + (0.433 + 0.900i)19-s + (−0.623 − 0.781i)20-s + (0.900 + 0.433i)22-s + (0.222 − 0.974i)23-s + ⋯
L(s,χ)  = 1  + (0.974 − 0.222i)2-s + (0.900 − 0.433i)4-s + (−0.222 − 0.974i)5-s + (−0.900 − 0.433i)7-s + (0.781 − 0.623i)8-s + (−0.433 − 0.900i)10-s + (0.781 + 0.623i)11-s + (−0.623 + 0.781i)13-s + (−0.974 − 0.222i)14-s + (0.623 − 0.781i)16-s + i·17-s + (0.433 + 0.900i)19-s + (−0.623 − 0.781i)20-s + (0.900 + 0.433i)22-s + (0.222 − 0.974i)23-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.653 - 0.757i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.653 - 0.757i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(87\)    =    \(3 \cdot 29\)
\( \varepsilon \)  =  $0.653 - 0.757i$
motivic weight  =  \(0\)
character  :  $\chi_{87} (14, \cdot )$
Sato-Tate  :  $\mu(28)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 87,\ (0:\ ),\ 0.653 - 0.757i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.421755791 - 0.6513343394i$
$L(\frac12,\chi)$  $\approx$  $1.421755791 - 0.6513343394i$
$L(\chi,1)$  $\approx$  1.515852990 - 0.4500496064i
$L(1,\chi)$  $\approx$  1.515852990 - 0.4500496064i

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

−30.89026172398750874494775550922, −29.742793875657619518929915012241, −29.25903499174205135062456765449, −27.519076565622611064780442134231, −26.31082661598456412128344109357, −25.35119678428309843541619662355, −24.431029211545119037423290038712, −23.03243733098151225684881066217, −22.32488662400290816585896809259, −21.67675365073721216477205350048, −19.99771609141464441824491696895, −19.19911887378523928454212084481, −17.68864723266802555300610930589, −16.218509719605149305172125816746, −15.3569082334792576304779636973, −14.31837270470587908452909627881, −13.23859370399916758783451173633, −11.953040253261293724822240752341, −11.012147489921689583595193797597, −9.44966914895003161058044106723, −7.52102882701628915212348136024, −6.574743851215106038782826367061, −5.38185497151867450100838436564, −3.54527762116227231774438990762, −2.71411951507047086895996879193, 1.680058947455952566698098077860, 3.66492229145882049030833676956, 4.608553182447082972319010892505, 6.12615541320523178926089779469, 7.33442938340726552147533383001, 9.19707778446097620071014503596, 10.4067920096107605338908308694, 12.09772707907527712811988299711, 12.577531439107180311650838311023, 13.84644044675218120637247718965, 14.9944223897229628148512182522, 16.34269154690480482507510727691, 16.94900454072427332517522125785, 19.16279486027204184152134600077, 19.91965201615822405811827190441, 20.82540970352698969528802332031, 22.08530589564730710724929133546, 22.99824430664483791647103225897, 24.033397634976995751827616436969, 24.894037138004055213298481323429, 26.04063018964394281476730281887, 27.61958631095917095647895629853, 28.74398701160786533537529161645, 29.363587569558026691042331963509, 30.71240359934784811776354312433

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