Properties

Label 1-23e2-529.98-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

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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: $\chi_{529} (98, \cdot )$
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 \( 1 + (-0.935 - 0.352i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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