Properties

Label 1-23e2-529.27-r0-0-0
Degree $1$
Conductor $529$
Sign $0.954 + 0.298i$
Analytic cond. $2.45666$
Root an. cond. $2.45666$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.984 − 0.172i)2-s + (−0.993 − 0.111i)3-s + (0.940 − 0.340i)4-s + (0.827 + 0.561i)5-s + (−0.998 + 0.0620i)6-s + (0.0806 + 0.996i)7-s + (0.867 − 0.498i)8-s + (0.975 + 0.221i)9-s + (0.912 + 0.409i)10-s + (−0.834 − 0.551i)11-s + (−0.972 + 0.233i)12-s + (0.988 + 0.148i)13-s + (0.251 + 0.967i)14-s + (−0.759 − 0.650i)15-s + (0.767 − 0.640i)16-s + (0.00620 − 0.999i)17-s + ⋯
L(s)  = 1  + (0.984 − 0.172i)2-s + (−0.993 − 0.111i)3-s + (0.940 − 0.340i)4-s + (0.827 + 0.561i)5-s + (−0.998 + 0.0620i)6-s + (0.0806 + 0.996i)7-s + (0.867 − 0.498i)8-s + (0.975 + 0.221i)9-s + (0.912 + 0.409i)10-s + (−0.834 − 0.551i)11-s + (−0.972 + 0.233i)12-s + (0.988 + 0.148i)13-s + (0.251 + 0.967i)14-s + (−0.759 − 0.650i)15-s + (0.767 − 0.640i)16-s + (0.00620 − 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.954 + 0.298i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.954 + 0.298i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.234230018 + 0.3415745663i\)
\(L(\frac12)\) \(\approx\) \(2.234230018 + 0.3415745663i\)
\(L(1)\) \(\approx\) \(1.687385006 + 0.06983286445i\)
\(L(1)\) \(\approx\) \(1.687385006 + 0.06983286445i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.984 + 0.172i)T \)
3 \( 1 + (0.993 + 0.111i)T \)
5 \( 1 + (-0.827 - 0.561i)T \)
7 \( 1 + (-0.0806 - 0.996i)T \)
11 \( 1 + (0.834 + 0.551i)T \)
13 \( 1 + (-0.988 - 0.148i)T \)
17 \( 1 + (-0.00620 + 0.999i)T \)
19 \( 1 + (0.514 - 0.857i)T \)
29 \( 1 + (-0.130 - 0.991i)T \)
31 \( 1 + (-0.179 - 0.983i)T \)
37 \( 1 + (0.743 - 0.668i)T \)
41 \( 1 + (-0.879 + 0.476i)T \)
43 \( 1 + (0.926 + 0.375i)T \)
47 \( 1 + (-0.682 + 0.730i)T \)
53 \( 1 + (-0.912 + 0.409i)T \)
59 \( 1 + (-0.922 + 0.386i)T \)
61 \( 1 + (-0.999 - 0.0248i)T \)
67 \( 1 + (0.426 - 0.904i)T \)
71 \( 1 + (0.709 + 0.704i)T \)
73 \( 1 + (-0.130 + 0.991i)T \)
79 \( 1 + (-0.323 - 0.946i)T \)
83 \( 1 + (0.117 + 0.993i)T \)
89 \( 1 + (0.791 + 0.611i)T \)
97 \( 1 + (0.935 - 0.352i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.451354338113048917638822214710, −22.7677449863433154721421281043, −21.79243738465068172680182165924, −20.98625295360591163573607499892, −20.65732162841060399107072099674, −19.434468454314814422357604193583, −17.941014116224827012850603581785, −17.33926026391743569883669629214, −16.662005120542910159060984029801, −15.82824033836602922342874636753, −15.01351421861630515709876575723, −13.680459830416936179610208116027, −13.12858802780591834211671049314, −12.58111753199970759526548426568, −11.30569904433457267631547973046, −10.6219831555545325763461068079, −9.90296554724356528922273824261, −8.28374378031321293915616015733, −7.188792878144190300766694780766, −6.24662280053851432218057872977, −5.59450757131466267504730187005, −4.575398690956679193929495343987, −3.980229912149146763368773631222, −2.28089268577894610249876921305, −1.11689642981428832003598770355, 1.46194623750701768626255900540, 2.47195097396267352792318559915, 3.5389162190579168156681538011, 5.089977452024580323269270504666, 5.54721162657638400712880749853, 6.32832851225613631546643643852, 7.12002299210783030473784425617, 8.61861584169951235117624590684, 10.07680932483945200368559187859, 10.70397986402379898936216269862, 11.51235561805878045767807342023, 12.33236034656926699331786781080, 13.22097292512526274463626815940, 13.917866618043664684396669479705, 14.9654952386148003027864778288, 15.93166776944007953012663334835, 16.4208996177415145818954426299, 17.78012655327315634540674219579, 18.51128241916968778154676469298, 19.01473937558972308662600564973, 20.74627044893338877847654335293, 21.24270981029957092393019569373, 21.90938789089360404484818861748, 22.627755706589569584187011644771, 23.344933567013983962537435160411

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