Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $0.668 - 0.743i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.642 − 0.766i)2-s + (−0.686 − 0.727i)3-s + (−0.173 + 0.984i)4-s + (−0.893 − 0.448i)5-s + (−0.116 + 0.993i)6-s + (−0.835 + 0.549i)7-s + (0.866 − 0.5i)8-s + (−0.0581 + 0.998i)9-s + (0.230 + 0.973i)10-s + (0.230 − 0.973i)11-s + (0.835 − 0.549i)12-s + (−0.549 − 0.835i)13-s + (0.957 + 0.286i)14-s + (0.286 + 0.957i)15-s + (−0.939 − 0.342i)16-s + (−0.342 + 0.939i)17-s + ⋯
L(s,χ)  = 1  + (−0.642 − 0.766i)2-s + (−0.686 − 0.727i)3-s + (−0.173 + 0.984i)4-s + (−0.893 − 0.448i)5-s + (−0.116 + 0.993i)6-s + (−0.835 + 0.549i)7-s + (0.866 − 0.5i)8-s + (−0.0581 + 0.998i)9-s + (0.230 + 0.973i)10-s + (0.230 − 0.973i)11-s + (0.835 − 0.549i)12-s + (−0.549 − 0.835i)13-s + (0.957 + 0.286i)14-s + (0.286 + 0.957i)15-s + (−0.939 − 0.342i)16-s + (−0.342 + 0.939i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.668 - 0.743i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.668 - 0.743i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $0.668 - 0.743i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (11, \cdot )$
Sato-Tate  :  $\mu(108)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (1:\ ),\ 0.668 - 0.743i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.3371689765 - 0.1503612997i$
$L(\frac12,\chi)$  $\approx$  $0.3371689765 - 0.1503612997i$
$L(\chi,1)$  $\approx$  0.3519299645 - 0.2588685398i
$L(1,\chi)$  $\approx$  0.3519299645 - 0.2588685398i

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

−18.36238491135534438587369524136, −17.53056543180055062869325936219, −16.83581383966959819522713336762, −16.39796804766929991293158197025, −15.84842254852439890047571900466, −15.11628335778832483864032292411, −14.71034262991048370089048220583, −13.891619586134129319171559230678, −12.89741635274521397314330578246, −11.84519779525631617562459102927, −11.53577115185241184504679833385, −10.54978008058328060362320722145, −10.04975984979671448759010384871, −9.41431142035902888944188548576, −8.88891292040645917306719028531, −7.590257569432735117321673711322, −7.05649800699326436084508678770, −6.78992361019883228062539156061, −5.7257793770944836542466058603, −4.98819644135611929368771172139, −4.14602046181389538654251285785, −3.7052102502481890293346918031, −2.438818780814837795406681502155, −1.13657880791892010863459322306, −0.19783561905689391743251654988, 0.462390140593531806576929031386, 0.99254151395313915710920651653, 2.17531881462532333668343132981, 2.9742845813169901207586171977, 3.627656616835353446296901272701, 4.68936109435722760631480734225, 5.42419691033448414450950317041, 6.42951360084668634055781179017, 7.06028560255843662706434747621, 7.89811144731342401744737831419, 8.50871011501050216149558465878, 9.03577250330854143297729834765, 10.04539160216804122791203892547, 10.849776550557974170239819194478, 11.32531460243451766562754219726, 12.04718403899697133525512182800, 12.59026050238869531187322872156, 13.04868963301112221707777732186, 13.65764767687256998864952047947, 15.05030845027025864078456182938, 15.695773720272237645647737080356, 16.584983338709628823990361835821, 16.76739000568878322997625244849, 17.65399196656444067266861418893, 18.32511323617684600785549043992

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