Properties

Degree 1
Conductor 311
Sign $-0.606 + 0.795i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.954 + 0.299i)2-s + (−0.954 − 0.299i)3-s + (0.820 − 0.571i)4-s + (−0.758 + 0.651i)5-s + 6-s + (−0.874 + 0.485i)7-s + (−0.612 + 0.790i)8-s + (0.820 + 0.571i)9-s + (0.528 − 0.848i)10-s + (−0.151 − 0.988i)11-s + (−0.954 + 0.299i)12-s + (0.820 + 0.571i)13-s + (0.688 − 0.724i)14-s + (0.918 − 0.394i)15-s + (0.347 − 0.937i)16-s + (−0.528 + 0.848i)17-s + ⋯
L(s,χ)  = 1  + (−0.954 + 0.299i)2-s + (−0.954 − 0.299i)3-s + (0.820 − 0.571i)4-s + (−0.758 + 0.651i)5-s + 6-s + (−0.874 + 0.485i)7-s + (−0.612 + 0.790i)8-s + (0.820 + 0.571i)9-s + (0.528 − 0.848i)10-s + (−0.151 − 0.988i)11-s + (−0.954 + 0.299i)12-s + (0.820 + 0.571i)13-s + (0.688 − 0.724i)14-s + (0.918 − 0.394i)15-s + (0.347 − 0.937i)16-s + (−0.528 + 0.848i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.606 + 0.795i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.606 + 0.795i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(311\)
\( \varepsilon \)  =  $-0.606 + 0.795i$
motivic weight  =  \(0\)
character  :  $\chi_{311} (228, \cdot )$
Sato-Tate  :  $\mu(62)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 311,\ (1:\ ),\ -0.606 + 0.795i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.1714132067 + 0.3463504552i$
$L(\frac12,\chi)$  $\approx$  $0.1714132067 + 0.3463504552i$
$L(\chi,1)$  $\approx$  0.4132084702 + 0.1034908339i
$L(1,\chi)$  $\approx$  0.4132084702 + 0.1034908339i

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

−24.82460236904407142816169039663, −23.6702908034825018825467952962, −22.96444780443214071100868730774, −22.084533935743456072156561829480, −20.75309004238098356182574094953, −20.22293794619588139731248330707, −19.37496200079123452075258908341, −18.137969306911378858779994826664, −17.58531586139576313611624257463, −16.43868133361082017164593484320, −15.97767323174415897253563229951, −15.30825399328722084300644757309, −13.07279499450020040247163377441, −12.571737163823900310450562180074, −11.44043225410318642141849625549, −10.8384230549621013328077135754, −9.65481037098545496575000339736, −9.06182082449756977348752430705, −7.45575774114904385636589839958, −6.98203683635694337268866465893, −5.519888259332949944180836370673, −4.23219149468859002695082961475, −3.17532124220374282851038683536, −1.1889315124519283425387859323, −0.27852711435939422442588696898, 0.873028989268912590299184005762, 2.50506600486463483491264716517, 3.92385975260159167762231502982, 5.87112878034377179107920079570, 6.23861187720922781682206645393, 7.297451072841415569405832364917, 8.28242507850255468791083047891, 9.42382655135846447591474619474, 10.674542625115862183249622548343, 11.18326212701151657170908149347, 12.06099711225947638806994904969, 13.24773613836857149093688891803, 14.69638600198447481292381436800, 15.858107095620966361365267349601, 16.180253954346625095108953212094, 17.14286501139143453691518156771, 18.42633075242402357725839515714, 18.800095392179036613937761106620, 19.35927925379338006869857883657, 20.75013091780063074027135199444, 22.01594328715890632827445379885, 22.749100622819712387827518744116, 23.75112259262925910261611687306, 24.31373988272568649803147718385, 25.44656989999947356542405476456

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