Properties

Degree 1
Conductor $ 3 \cdot 97 $
Sign $0.977 - 0.213i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.382 − 0.923i)2-s + (−0.707 − 0.707i)4-s + (0.555 + 0.831i)5-s + (−0.555 + 0.831i)7-s + (−0.923 + 0.382i)8-s + (0.980 − 0.195i)10-s + (0.923 − 0.382i)11-s + (−0.831 + 0.555i)13-s + (0.555 + 0.831i)14-s + i·16-s + (0.831 − 0.555i)17-s + (0.555 + 0.831i)19-s + (0.195 − 0.980i)20-s i·22-s + (0.980 + 0.195i)23-s + ⋯
L(s,χ)  = 1  + (0.382 − 0.923i)2-s + (−0.707 − 0.707i)4-s + (0.555 + 0.831i)5-s + (−0.555 + 0.831i)7-s + (−0.923 + 0.382i)8-s + (0.980 − 0.195i)10-s + (0.923 − 0.382i)11-s + (−0.831 + 0.555i)13-s + (0.555 + 0.831i)14-s + i·16-s + (0.831 − 0.555i)17-s + (0.555 + 0.831i)19-s + (0.195 − 0.980i)20-s i·22-s + (0.980 + 0.195i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 291 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.977 - 0.213i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 291 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.977 - 0.213i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(291\)    =    \(3 \cdot 97\)
\( \varepsilon \)  =  $0.977 - 0.213i$
motivic weight  =  \(0\)
character  :  $\chi_{291} (245, \cdot )$
Sato-Tate  :  $\mu(32)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 291,\ (0:\ ),\ 0.977 - 0.213i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.433141829 - 0.1545574920i$
$L(\frac12,\chi)$  $\approx$  $1.433141829 - 0.1545574920i$
$L(\chi,1)$  $\approx$  1.218762978 - 0.2700774730i
$L(1,\chi)$  $\approx$  1.218762978 - 0.2700774730i

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

−25.3614695056156557477906559314, −24.64373556679611065973608824804, −23.8592718432157294650476009627, −22.852675990910609413795105267524, −22.165816469683735288084342405932, −21.14010433832777971901286311556, −20.16332651635887080590500957699, −19.22904421935057890377247677218, −17.6729700847010653838792453983, −17.10786508094331787835054751272, −16.5549052659579875417475167543, −15.40162358525435442000119620065, −14.42209264760450482536455830074, −13.50702855762997852391234294449, −12.7857937952693601717941877939, −11.90984472235887036040473715168, −10.05617364785452785647137598788, −9.41605706625229491424898726156, −8.273382782004097562629628727402, −7.179348963195987084661128617504, −6.30718302977139717278628559232, −5.13222839700954711629306084745, −4.30963728726370970645911649454, −3.02720824845447158551064026805, −0.96506187386214190054966696277, 1.51954529089152718011368222893, 2.776176292676472124844526037551, 3.455487421373206314884868818048, 5.049222143180489619887598528686, 5.99583090665308534878743942082, 6.99146430029407578656811136206, 8.82018763761197295915697855647, 9.633929904774061610528280014358, 10.353630414221284935316307295490, 11.70474937526733155377139309031, 12.12449169060772942517225583493, 13.41825693296510597702553854419, 14.3410539945355273479340675987, 14.843707723023554238336502639075, 16.28114539832967930023142492879, 17.50390907852343062580579668944, 18.54583836464970069016265135266, 19.07592245543681270107542496355, 19.90210643706382878491348096933, 21.406311165406616323309941081204, 21.60464491442570165327265836531, 22.63433715554914563227641854400, 23.16849372447738363564890277227, 24.73113930292178258867000009399, 25.21526362124552340806882141044

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