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Properties

Label 1-503-503.2-r0-0-0
Degree $1$
Conductor $503$
Sign $-0.944 - 0.327i$
Analytic cond. $2.33592$
Root an. cond. $2.33592$
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  + (−0.205 + 0.978i)2-s + (0.143 − 0.989i)3-s + (−0.915 − 0.401i)4-s + (−0.817 + 0.575i)5-s + (0.939 + 0.343i)6-s + (−0.787 − 0.615i)7-s + (0.580 − 0.814i)8-s + (−0.958 − 0.283i)9-s + (−0.395 − 0.918i)10-s + (0.810 − 0.585i)11-s + (−0.528 + 0.848i)12-s + (0.795 − 0.605i)13-s + (0.764 − 0.644i)14-s + (0.452 + 0.891i)15-s + (0.677 + 0.735i)16-s + (0.600 + 0.799i)17-s + ⋯
L(s)  = 1  + (−0.205 + 0.978i)2-s + (0.143 − 0.989i)3-s + (−0.915 − 0.401i)4-s + (−0.817 + 0.575i)5-s + (0.939 + 0.343i)6-s + (−0.787 − 0.615i)7-s + (0.580 − 0.814i)8-s + (−0.958 − 0.283i)9-s + (−0.395 − 0.918i)10-s + (0.810 − 0.585i)11-s + (−0.528 + 0.848i)12-s + (0.795 − 0.605i)13-s + (0.764 − 0.644i)14-s + (0.452 + 0.891i)15-s + (0.677 + 0.735i)16-s + (0.600 + 0.799i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(503\)
Sign: $-0.944 - 0.327i$
Analytic conductor: \(2.33592\)
Root analytic conductor: \(2.33592\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{503} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 503,\ (0:\ ),\ -0.944 - 0.327i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01902766215 - 0.1129242956i\)
\(L(\frac12)\) \(\approx\) \(0.01902766215 - 0.1129242956i\)
\(L(1)\) \(\approx\) \(0.5849606415 + 0.03141140566i\)
\(L(1)\) \(\approx\) \(0.5849606415 + 0.03141140566i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad503 \( 1 \)
good2 \( 1 + (-0.205 + 0.978i)T \)
3 \( 1 + (0.143 - 0.989i)T \)
5 \( 1 + (-0.817 + 0.575i)T \)
7 \( 1 + (-0.787 - 0.615i)T \)
11 \( 1 + (0.810 - 0.585i)T \)
13 \( 1 + (0.795 - 0.605i)T \)
17 \( 1 + (0.600 + 0.799i)T \)
19 \( 1 + (-0.668 + 0.743i)T \)
23 \( 1 + (-0.845 - 0.533i)T \)
29 \( 1 + (-0.971 - 0.235i)T \)
31 \( 1 + (-0.986 + 0.161i)T \)
37 \( 1 + (-0.229 + 0.973i)T \)
41 \( 1 + (-0.925 - 0.378i)T \)
43 \( 1 + (-0.934 - 0.355i)T \)
47 \( 1 + (0.997 - 0.0750i)T \)
53 \( 1 + (-0.668 - 0.743i)T \)
59 \( 1 + (-0.704 - 0.709i)T \)
61 \( 1 + (-0.463 + 0.886i)T \)
67 \( 1 + (0.452 - 0.891i)T \)
71 \( 1 + (-0.951 + 0.307i)T \)
73 \( 1 + (0.0437 + 0.999i)T \)
79 \( 1 + (-0.993 + 0.112i)T \)
83 \( 1 + (-0.570 + 0.821i)T \)
89 \( 1 + (-0.871 + 0.490i)T \)
97 \( 1 + (-0.894 + 0.446i)T \)
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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

−23.60480262765266679374306045606, −22.94530547971743071574833612988, −22.0979034594572018727615882996, −21.49490869988713037603845190949, −20.440691466356292207937373644107, −19.99853548046190197295872690133, −19.20751519718363040955833848648, −18.35747546414714279648273303109, −17.03882429607636869780193931453, −16.4007586114252207346162343682, −15.57488569905543906724334087194, −14.61240502903914228936185010239, −13.53053505343046167697699677963, −12.494899935009884686335594295076, −11.72221181113309918874254557165, −11.100179430102625800573128516453, −9.86032463761562317982511972351, −9.12037719017512711926411152899, −8.74531304097538647786964164208, −7.40759949317825718661890655046, −5.77195738644168768289691909542, −4.65416967196536776144304469007, −3.87053320183407618124571494860, −3.17584555705518893445004347686, −1.77760614682752651685497946797, 0.06893464748424861400972464235, 1.42432125976602438245905522612, 3.43676589879832811063442486364, 3.87497193286476432151593632545, 5.82384853456366690645423193977, 6.39589926709444533991417684451, 7.17632512634044851581857792833, 8.10219739486292166439713148178, 8.63576588521183928566681933507, 10.059451839296470723444161438570, 10.958429536736967904992745503849, 12.23426563965199169623198169888, 13.03865971465879425359810859667, 13.956769456527446839263545984780, 14.62272525462618031638770481278, 15.52927368719840581626766081603, 16.64406849835263305759730984951, 17.06608594605033607783227858809, 18.44192199979280453078816743327, 18.777502339242834032919252975693, 19.57238172823300139276529452202, 20.31506290915143260213427741475, 22.15960016780567237508697605009, 22.685629571500258958735759473838, 23.60872568789409154212050254183

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