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Properties

Label 1-23e2-529.427-r0-0-0
Degree $1$
Conductor $529$
Sign $-0.894 - 0.446i$
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  + (0.524 − 0.851i)2-s + (−0.0186 − 0.999i)3-s + (−0.449 − 0.893i)4-s + (0.995 − 0.0991i)5-s + (−0.860 − 0.508i)6-s + (0.969 − 0.245i)7-s + (−0.996 − 0.0868i)8-s + (−0.999 + 0.0372i)9-s + (0.437 − 0.899i)10-s + (0.813 − 0.581i)11-s + (−0.885 + 0.465i)12-s + (0.999 − 0.0248i)13-s + (0.299 − 0.954i)14-s + (−0.117 − 0.993i)15-s + (−0.596 + 0.802i)16-s + (−0.966 − 0.257i)17-s + ⋯
L(s)  = 1  + (0.524 − 0.851i)2-s + (−0.0186 − 0.999i)3-s + (−0.449 − 0.893i)4-s + (0.995 − 0.0991i)5-s + (−0.860 − 0.508i)6-s + (0.969 − 0.245i)7-s + (−0.996 − 0.0868i)8-s + (−0.999 + 0.0372i)9-s + (0.437 − 0.899i)10-s + (0.813 − 0.581i)11-s + (−0.885 + 0.465i)12-s + (0.999 − 0.0248i)13-s + (0.299 − 0.954i)14-s + (−0.117 − 0.993i)15-s + (−0.596 + 0.802i)16-s + (−0.966 − 0.257i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $-0.894 - 0.446i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (427, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ -0.894 - 0.446i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5168624632 - 2.194203035i\)
\(L(\frac12)\) \(\approx\) \(0.5168624632 - 2.194203035i\)
\(L(1)\) \(\approx\) \(1.035280812 - 1.305415454i\)
\(L(1)\) \(\approx\) \(1.035280812 - 1.305415454i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.524 - 0.851i)T \)
3 \( 1 + (-0.0186 - 0.999i)T \)
5 \( 1 + (0.995 - 0.0991i)T \)
7 \( 1 + (0.969 - 0.245i)T \)
11 \( 1 + (0.813 - 0.581i)T \)
13 \( 1 + (0.999 - 0.0248i)T \)
17 \( 1 + (-0.966 - 0.257i)T \)
19 \( 1 + (0.767 + 0.640i)T \)
29 \( 1 + (-0.691 - 0.722i)T \)
31 \( 1 + (-0.287 + 0.957i)T \)
37 \( 1 + (0.798 + 0.601i)T \)
41 \( 1 + (-0.426 - 0.904i)T \)
43 \( 1 + (-0.896 - 0.443i)T \)
47 \( 1 + (-0.990 - 0.136i)T \)
53 \( 1 + (0.437 + 0.899i)T \)
59 \( 1 + (-0.556 + 0.831i)T \)
61 \( 1 + (0.503 + 0.863i)T \)
67 \( 1 + (-0.944 + 0.329i)T \)
71 \( 1 + (0.130 + 0.991i)T \)
73 \( 1 + (-0.691 + 0.722i)T \)
79 \( 1 + (-0.311 + 0.950i)T \)
83 \( 1 + (-0.239 - 0.970i)T \)
89 \( 1 + (-0.806 + 0.591i)T \)
97 \( 1 + (-0.834 - 0.551i)T \)
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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

−23.909143725705985663624270872713, −22.823876990730446199163403469511, −22.07073539309404275558759827242, −21.62289793642030252212315610724, −20.74803525605641757183700543724, −20.1136151007166580354361468871, −18.17699536446690681550277302224, −17.80048184457085977647240066785, −16.94336996209377307343231716586, −16.19247814129272682220899677447, −15.06843427925726259844654544165, −14.72308258562145661924686101163, −13.80382196362751945634968446936, −13.03729829371201719170254456598, −11.60368266668201005661707152734, −11.01216541484498605677093458636, −9.559717746353223807927524207206, −9.04403014545222227049869934701, −8.120554089983510422612028326131, −6.74353603734127894103345588307, −5.91329425093774447614484486491, −5.03105408932225159164798424947, −4.32171774644274911439822473974, −3.19849881700787782487416643220, −1.84653587917472892515507732485, 1.1918023942284580955610764159, 1.637725745006692214386113309780, 2.79296162808625456923374867131, 4.01946250147996987419363589582, 5.3270268820179992892212268956, 5.99096562872338053764774049925, 6.92941557460343531242962335235, 8.44558127655038258883098018715, 9.04362086303675053290931750502, 10.325369732056570353264887945738, 11.306874415244948292560589821356, 11.771361790530425102513497829, 12.969450047416413315084519803432, 13.73173722287857871972215730958, 14.01690515686803027609339812907, 14.99751275634582475236304806392, 16.59489300749855835727159078895, 17.59625423817833741472977203765, 18.19384034923082572068385626648, 18.817750087926951776242713281266, 20.03759480242335376358382640122, 20.5057758752795199544373005441, 21.41928600887352525569329008296, 22.249611284189150230005173595966, 23.05450636457613464905223266580

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