Properties

Degree 1
Conductor 89
Sign $0.656 + 0.754i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.654 + 0.755i)2-s + (−0.989 − 0.142i)3-s + (−0.142 − 0.989i)4-s + (−0.841 − 0.540i)5-s + (0.755 − 0.654i)6-s + (−0.540 + 0.841i)7-s + (0.841 + 0.540i)8-s + (0.959 + 0.281i)9-s + (0.959 − 0.281i)10-s + (0.841 − 0.540i)11-s + i·12-s + (0.989 + 0.142i)13-s + (−0.281 − 0.959i)14-s + (0.755 + 0.654i)15-s + (−0.959 + 0.281i)16-s + (0.654 + 0.755i)17-s + ⋯
L(s,χ)  = 1  + (−0.654 + 0.755i)2-s + (−0.989 − 0.142i)3-s + (−0.142 − 0.989i)4-s + (−0.841 − 0.540i)5-s + (0.755 − 0.654i)6-s + (−0.540 + 0.841i)7-s + (0.841 + 0.540i)8-s + (0.959 + 0.281i)9-s + (0.959 − 0.281i)10-s + (0.841 − 0.540i)11-s + i·12-s + (0.989 + 0.142i)13-s + (−0.281 − 0.959i)14-s + (0.755 + 0.654i)15-s + (−0.959 + 0.281i)16-s + (0.654 + 0.755i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.656 + 0.754i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.656 + 0.754i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(89\)
\( \varepsilon \)  =  $0.656 + 0.754i$
motivic weight  =  \(0\)
character  :  $\chi_{89} (80, \cdot )$
Sato-Tate  :  $\mu(44)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 89,\ (0:\ ),\ 0.656 + 0.754i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.4158444553 + 0.1894103830i$
$L(\frac12,\chi)$  $\approx$  $0.4158444553 + 0.1894103830i$
$L(\chi,1)$  $\approx$  0.5144323667 + 0.1472591037i
$L(1,\chi)$  $\approx$  0.5144323667 + 0.1472591037i

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

−30.01217888754078127329781047383, −29.454520926116442819088309483730, −28.01795443156340097016638756770, −27.571360189783843521789737610553, −26.53195548342700551883368913085, −25.49792228515345307695624050276, −23.55931009633606126122163018606, −22.841386912615525302033955090195, −21.998999717603721236306952066229, −20.58578950502549706198600498434, −19.5721565557788679370469044004, −18.56483802829128814991844504316, −17.49093299774479306774697367885, −16.51780393518307985036544140828, −15.52390367879574164593921219856, −13.54124501063843880880210493092, −12.22530044814754034977579853369, −11.326199425166666237972404349041, −10.50204584435196300675773573791, −9.33736709335113411198878498983, −7.51066171171937070883225793348, −6.6585216375215555994054288618, −4.39083255029108235376156378565, −3.37567813689295228204019526845, −0.93454142619470117924016011273, 1.14690964757162530892452244003, 4.14139494603407596395670769470, 5.73202826160267484691900857108, 6.464282544046523496442809289933, 8.0498314482931620606361554175, 9.053455234082121863438634909584, 10.56039320978762682643093339673, 11.75041611270494197991207859514, 12.8078684028718107841042487254, 14.61602841855421078082112963030, 16.05208822599465299468322878401, 16.30291932546810401527341501340, 17.546860419866277577080019537884, 18.89136044179471929182023582221, 19.28845131837075334139345554624, 21.13477403428199345203121119688, 22.68689232662207073919294491474, 23.30446887207431414078268710933, 24.44027107849386427959744541547, 25.15557129548040364184343529705, 26.6627391701024570174219480508, 27.74246483334562323795446624972, 28.21796986909471296285377658511, 29.16065343811021163890642678214, 30.657245490362584482923698336011

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