Properties

Degree 1
Conductor $ 13 \cdot 89 $
Sign $-0.605 + 0.795i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.618 − 0.786i)2-s + (−0.520 − 0.853i)3-s + (−0.235 − 0.971i)4-s + (−0.415 + 0.909i)5-s + (−0.992 − 0.118i)6-s + (0.771 + 0.636i)7-s + (−0.909 − 0.415i)8-s + (−0.458 + 0.888i)9-s + (0.458 + 0.888i)10-s + (0.0950 − 0.995i)11-s + (−0.707 + 0.707i)12-s + (0.977 − 0.212i)14-s + (0.992 − 0.118i)15-s + (−0.888 + 0.458i)16-s + (0.618 + 0.786i)17-s + (0.415 + 0.909i)18-s + ⋯
L(s,χ)  = 1  + (0.618 − 0.786i)2-s + (−0.520 − 0.853i)3-s + (−0.235 − 0.971i)4-s + (−0.415 + 0.909i)5-s + (−0.992 − 0.118i)6-s + (0.771 + 0.636i)7-s + (−0.909 − 0.415i)8-s + (−0.458 + 0.888i)9-s + (0.458 + 0.888i)10-s + (0.0950 − 0.995i)11-s + (−0.707 + 0.707i)12-s + (0.977 − 0.212i)14-s + (0.992 − 0.118i)15-s + (−0.888 + 0.458i)16-s + (0.618 + 0.786i)17-s + (0.415 + 0.909i)18-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.605 + 0.795i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.605 + 0.795i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1157\)    =    \(13 \cdot 89\)
\( \varepsilon \)  =  $-0.605 + 0.795i$
motivic weight  =  \(0\)
character  :  $\chi_{1157} (310, \cdot )$
Sato-Tate  :  $\mu(264)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 1157,\ (0:\ ),\ -0.605 + 0.795i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.1998447002 - 0.4034252527i$
$L(\frac12,\chi)$  $\approx$  $-0.1998447002 - 0.4034252527i$
$L(\chi,1)$  $\approx$  0.7143357083 - 0.5457298236i
$L(1,\chi)$  $\approx$  0.7143357083 - 0.5457298236i

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

−21.75841506776597211273774885605, −21.14610393677389187232325783696, −20.38339186214227392286041245734, −19.99929290851287238877464873265, −18.260863409572509346413997436561, −17.560471941309428560136355726393, −16.87879725562356013545832567302, −16.4363878182557011522245531165, −15.54846010944395200803562133942, −14.95351180217256755818452739037, −14.217351562247636537405939868062, −13.23655867910032393392950727557, −12.29720032245959963002213561023, −11.79127226272493023086900157378, −10.90100454104416834747121101207, −9.75685986702115330251927334315, −9.002597244102272520909178666051, −8.05807320980136980584503223386, −7.34662480688759034218574564786, −6.34089169252630716013310346225, −5.165762294935666556851269901696, −4.80952214037434501654803774900, −4.14383181334581950589648021411, −3.27257368283518699752345858545, −1.53568026597019196125979714174, 0.152727053609296844839855283325, 1.65836291215662961759616916462, 2.28273341524059452492626168008, 3.331059431990759067313225188094, 4.26314293537035998633265750466, 5.503225340458374699563557652394, 6.00819853594357721423077668737, 6.81084808445984341699586472771, 8.06367601710325018184830279791, 8.590126877573993196931052620617, 10.15431577616300961036935023883, 10.845556234597045728949690001599, 11.4022937691955449206037139627, 12.09072841846364954937563621402, 12.71679904647893200002264213089, 13.77312833489963754829330516667, 14.40157890441786466571507350483, 14.9790959020001543142539014415, 16.027050495124324185650688669275, 17.13874753347587150664882684879, 18.077733412906150412625632383460, 18.68723279337206518478403004895, 19.09619360317226148860267874552, 19.80874730785455277657529762549, 20.97625552201258975976124294126

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