Properties

Degree 1
Conductor $ 2^{7} \cdot 5 $
Sign $0.0136 - 0.999i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.555 − 0.831i)3-s + (0.923 + 0.382i)7-s + (−0.382 − 0.923i)9-s + (−0.555 − 0.831i)11-s + (−0.195 − 0.980i)13-s + (0.707 + 0.707i)17-s + (0.195 + 0.980i)19-s + (0.831 − 0.555i)21-s + (−0.382 − 0.923i)23-s + (−0.980 − 0.195i)27-s + (0.555 − 0.831i)29-s i·31-s − 33-s + (−0.980 − 0.195i)37-s + (−0.923 − 0.382i)39-s + ⋯
L(s,χ)  = 1  + (0.555 − 0.831i)3-s + (0.923 + 0.382i)7-s + (−0.382 − 0.923i)9-s + (−0.555 − 0.831i)11-s + (−0.195 − 0.980i)13-s + (0.707 + 0.707i)17-s + (0.195 + 0.980i)19-s + (0.831 − 0.555i)21-s + (−0.382 − 0.923i)23-s + (−0.980 − 0.195i)27-s + (0.555 − 0.831i)29-s i·31-s − 33-s + (−0.980 − 0.195i)37-s + (−0.923 − 0.382i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.0136 - 0.999i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 640 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.0136 - 0.999i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(640\)    =    \(2^{7} \cdot 5\)
\( \varepsilon \)  =  $0.0136 - 0.999i$
motivic weight  =  \(0\)
character  :  $\chi_{640} (187, \cdot )$
Sato-Tate  :  $\mu(32)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 640,\ (0:\ ),\ 0.0136 - 0.999i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.238313739 - 1.221570131i$
$L(\frac12,\chi)$  $\approx$  $1.238313739 - 1.221570131i$
$L(\chi,1)$  $\approx$  1.220739478 - 0.5182415307i
$L(1,\chi)$  $\approx$  1.220739478 - 0.5182415307i

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

−23.22431916753504336946878169922, −22.03126329889874144259568391219, −21.370809109975608791923826046604, −20.71864160821592185611155121628, −20.00594557108719578776604876805, −19.202140576169568952485294061084, −18.01901628878706827559126052345, −17.37923128118155378644828172334, −16.27440289356125235974576481310, −15.695955489053998945575005471318, −14.63938948623417209537485622663, −14.16699706850828269102469135071, −13.3119785234936332813411340258, −11.98463456005145347368256789750, −11.20984286882469426536853647731, −10.27867863519941538916193968141, −9.535600152037778102254692937919, −8.648095437839982545880868667423, −7.671243316379763737413933662782, −6.956343855318058871265144859003, −5.140799096270448118301797071208, −4.8462816248817088762344233509, −3.73888595617254958148584144206, −2.6177197467169889042464617751, −1.56469091222120936281672538420, 0.83840261009821719814935236802, 2.036496053035344318161834371406, 2.91963353192088124571084981411, 4.02376878257563000487307221185, 5.53317023187544418294266955431, 6.023430814647700274040365097795, 7.48554308046294744101043902366, 8.13464473204906415630521815874, 8.59447983609936436919540814388, 9.97723544152017323672394467895, 10.87208619973170439868172324813, 11.99924562962869604743818274270, 12.554445520504506412576255399181, 13.56046301539026451733282728816, 14.3503602158802125819039259158, 14.97832928826983636241958189580, 15.961565265805326085281375836179, 17.194340452905873917915526665215, 17.82637760056905496700744877005, 18.81724929346644565208557643903, 19.07282936329262517859384718311, 20.51374502623318716142397913927, 20.733614439548969691690332059983, 21.7947990115335676134320350545, 22.856162129325108447141047953894

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