Properties

Degree 1
Conductor 113
Sign $0.895 - 0.445i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.900 + 0.433i)2-s + (0.532 − 0.846i)3-s + (0.623 + 0.781i)4-s + (−0.532 − 0.846i)5-s + (0.846 − 0.532i)6-s + (−0.222 − 0.974i)7-s + (0.222 + 0.974i)8-s + (−0.433 − 0.900i)9-s + (−0.111 − 0.993i)10-s + (0.781 + 0.623i)11-s + (0.993 − 0.111i)12-s + (−0.974 + 0.222i)13-s + (0.222 − 0.974i)14-s − 15-s + (−0.222 + 0.974i)16-s + (0.330 + 0.943i)17-s + ⋯
L(s,χ)  = 1  + (0.900 + 0.433i)2-s + (0.532 − 0.846i)3-s + (0.623 + 0.781i)4-s + (−0.532 − 0.846i)5-s + (0.846 − 0.532i)6-s + (−0.222 − 0.974i)7-s + (0.222 + 0.974i)8-s + (−0.433 − 0.900i)9-s + (−0.111 − 0.993i)10-s + (0.781 + 0.623i)11-s + (0.993 − 0.111i)12-s + (−0.974 + 0.222i)13-s + (0.222 − 0.974i)14-s − 15-s + (−0.222 + 0.974i)16-s + (0.330 + 0.943i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.895 - 0.445i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.895 - 0.445i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(113\)
\( \varepsilon \)  =  $0.895 - 0.445i$
motivic weight  =  \(0\)
character  :  $\chi_{113} (41, \cdot )$
Sato-Tate  :  $\mu(56)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 113,\ (0:\ ),\ 0.895 - 0.445i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.745960617 - 0.4102378911i$
$L(\frac12,\chi)$  $\approx$  $1.745960617 - 0.4102378911i$
$L(\chi,1)$  $\approx$  1.704661986 - 0.2109214173i
$L(1,\chi)$  $\approx$  1.704661986 - 0.2109214173i

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

−29.7318847658319795507308487262, −28.40411007145050838003737667325, −27.44949933410870229698631190284, −26.52936115100747522403007118817, −25.183214790692672083857861825524, −24.42584602745893441531739093406, −22.78557817953917359505364076781, −22.13237554242597298405652953569, −21.58697043794398374091709116483, −20.13057733345704556507314351284, −19.46733052310782078012461252906, −18.432620299640501498827745041301, −16.30571427255691892701838818180, −15.51101144130526194421871935928, −14.582234937245780123001867642040, −13.93299491115363073860035567842, −12.16943779294809252481842993681, −11.389487145902587723909955692425, −10.177118118878151471104587482643, −9.11989009572641941778798020282, −7.39160177216218007457360005780, −5.89265371301575833961767454505, −4.62687105505777762443321133192, −3.26213256516535542820509257825, −2.59427360310266373543812548985, 1.64593355434835694304977746633, 3.51637424427645510144227149505, 4.48117973251942682476224371323, 6.17409796904584843000023201290, 7.414939989857515749276221879032, 8.02237213086416954933032992720, 9.64504043692329226677812451708, 11.78610140037913872645859882752, 12.37111600919757615096036397902, 13.44427054606020397722521642447, 14.322395276128580374424746200430, 15.396114822858175509905560572179, 16.82030320152874390589628859218, 17.38345127985090255657123326010, 19.39342129468815187330100544076, 20.03790478981082593717767134475, 20.85792842392962253087819611728, 22.44972852435178886202527694282, 23.416432632977247464197594694709, 24.17345383290683974171642886446, 24.87994737895012332581494148843, 25.971045702247032381171583986329, 26.951993847600067371845905801114, 28.575105340044938136224018731730, 29.710061916274890018996990007294

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