Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $0.429 + 0.903i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.766 − 0.642i)2-s + (0.396 − 0.918i)3-s + (0.173 + 0.984i)4-s + (−0.973 − 0.230i)5-s + (−0.893 + 0.448i)6-s + (−0.286 − 0.957i)7-s + (0.5 − 0.866i)8-s + (−0.686 − 0.727i)9-s + (0.597 + 0.802i)10-s + (0.597 − 0.802i)11-s + (0.973 + 0.230i)12-s + (0.286 + 0.957i)13-s + (−0.396 + 0.918i)14-s + (−0.597 + 0.802i)15-s + (−0.939 + 0.342i)16-s + (0.939 − 0.342i)17-s + ⋯
L(s,χ)  = 1  + (−0.766 − 0.642i)2-s + (0.396 − 0.918i)3-s + (0.173 + 0.984i)4-s + (−0.973 − 0.230i)5-s + (−0.893 + 0.448i)6-s + (−0.286 − 0.957i)7-s + (0.5 − 0.866i)8-s + (−0.686 − 0.727i)9-s + (0.597 + 0.802i)10-s + (0.597 − 0.802i)11-s + (0.973 + 0.230i)12-s + (0.286 + 0.957i)13-s + (−0.396 + 0.918i)14-s + (−0.597 + 0.802i)15-s + (−0.939 + 0.342i)16-s + (0.939 − 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.429 + 0.903i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.429 + 0.903i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $0.429 + 0.903i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (3896, \cdot )$
Sato-Tate  :  $\mu(54)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ 0.429 + 0.903i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.1394903302 - 0.08812613310i$
$L(\frac12,\chi)$  $\approx$  $-0.1394903302 - 0.08812613310i$
$L(\chi,1)$  $\approx$  0.4454096113 - 0.4286718081i
$L(1,\chi)$  $\approx$  0.4454096113 - 0.4286718081i

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

−18.92017850413091293991956108158, −18.576526659063810559719706149870, −17.47172286241732340805145206389, −16.91398467319231868062198104322, −16.17263079139900336207551488413, −15.5760328277488630551861019596, −15.08297405141509744841439477116, −14.73309235558031922886580996381, −14.03830270932043498342639495443, −12.659531888629468138710191352745, −12.175135641783413842747438431201, −11.162756121606189743530313939947, −10.57885402142086100084297656222, −9.985265498916379265083588737893, −9.182224245918526478566582751051, −8.59412155272961708798418095228, −8.070858610507056931293883367623, −7.34373361448522105625618117292, −6.43541943727658797671986258706, −5.64292509996866017786261395741, −4.960934387100053667334491223321, −4.09706918462936970666566553297, −3.258481367470238544878355276843, −2.49555629862343808873804866014, −1.39635376399673968219765052094, 0.06653216340182168602468243904, 1.05425119231757106778739250995, 1.44039030579767611599897986075, 2.77060413182123014319840917359, 3.410751401617605777861244213220, 3.921002316061739675100186250424, 4.879611241412542261228081264100, 6.54825081159790460683463003390, 6.725095884514478683681273619341, 7.64862402457652023447944494633, 8.07428780548086767008871380472, 8.990521772793471658214545840945, 9.23644237910337799259459170185, 10.44362077066017987662677250623, 11.19775759503385158927110465062, 11.64820417207093012014217716970, 12.27958362613874468688058796148, 13.08318514920772620549839127725, 13.571989228023736280404592222232, 14.346753297953880307176267543359, 15.18938442345004667362350481914, 16.3538179861650914124066689573, 16.597976715447125014218634944076, 17.20914925815442738501683689705, 18.17250942654185556283805203150

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