↑

Properties

Label 1-1021-1021.10-r1-0-0
Degree $1$
Conductor $1021$
Sign $0.924 - 0.382i$
Analytic cond. $109.721$
Root an. cond. $109.721$
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  + (−0.462 + 0.886i)2-s + (−0.0922 + 0.995i)3-s + (−0.572 − 0.819i)4-s + (−0.456 − 0.889i)5-s + (−0.840 − 0.542i)6-s + (0.0738 + 0.997i)7-s + (0.991 − 0.128i)8-s + (−0.982 − 0.183i)9-s + (0.999 + 0.00615i)10-s + (0.995 + 0.0984i)11-s + (0.869 − 0.494i)12-s + (−0.895 − 0.445i)13-s + (−0.918 − 0.395i)14-s + (0.927 − 0.372i)15-s + (−0.343 + 0.938i)16-s + (0.836 − 0.547i)17-s + ⋯
L(s)  = 1  + (−0.462 + 0.886i)2-s + (−0.0922 + 0.995i)3-s + (−0.572 − 0.819i)4-s + (−0.456 − 0.889i)5-s + (−0.840 − 0.542i)6-s + (0.0738 + 0.997i)7-s + (0.991 − 0.128i)8-s + (−0.982 − 0.183i)9-s + (0.999 + 0.00615i)10-s + (0.995 + 0.0984i)11-s + (0.869 − 0.494i)12-s + (−0.895 − 0.445i)13-s + (−0.918 − 0.395i)14-s + (0.927 − 0.372i)15-s + (−0.343 + 0.938i)16-s + (0.836 − 0.547i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.924 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.924 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $0.924 - 0.382i$
Analytic conductor: \(109.721\)
Root analytic conductor: \(109.721\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (1:\ ),\ 0.924 - 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4033678738 - 0.08015219911i\)
\(L(\frac12)\) \(\approx\) \(0.4033678738 - 0.08015219911i\)
\(L(1)\) \(\approx\) \(0.5270501105 + 0.3576095019i\)
\(L(1)\) \(\approx\) \(0.5270501105 + 0.3576095019i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (-0.462 + 0.886i)T \)
3 \( 1 + (-0.0922 + 0.995i)T \)
5 \( 1 + (-0.456 - 0.889i)T \)
7 \( 1 + (0.0738 + 0.997i)T \)
11 \( 1 + (0.995 + 0.0984i)T \)
13 \( 1 + (-0.895 - 0.445i)T \)
17 \( 1 + (0.836 - 0.547i)T \)
19 \( 1 + (-0.943 - 0.332i)T \)
23 \( 1 + (-0.641 + 0.767i)T \)
29 \( 1 + (0.434 + 0.900i)T \)
31 \( 1 + (0.958 + 0.285i)T \)
37 \( 1 + (0.954 + 0.297i)T \)
41 \( 1 + (-0.332 + 0.943i)T \)
43 \( 1 + (0.135 - 0.990i)T \)
47 \( 1 + (0.116 + 0.993i)T \)
53 \( 1 + (-0.997 - 0.0677i)T \)
59 \( 1 + (-0.947 + 0.320i)T \)
61 \( 1 + (-0.801 + 0.597i)T \)
67 \( 1 + (0.669 - 0.743i)T \)
71 \( 1 + (-0.128 - 0.991i)T \)
73 \( 1 + (-0.918 + 0.395i)T \)
79 \( 1 + (0.993 + 0.110i)T \)
83 \( 1 + (0.908 - 0.417i)T \)
89 \( 1 + (-0.389 + 0.920i)T \)
97 \( 1 + (-0.473 - 0.881i)T \)
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

−21.46995353319834959582374798821, −20.34024182699378383815868177970, −19.65923254876785625259985252481, −19.13134550392434437885017885657, −18.68542482104506109179736829205, −17.52533466928852084672932170927, −17.15225707693499559565406355549, −16.41978328812844976096623044625, −14.689835479148107963480617713509, −14.24765064645311828858617407724, −13.483849595794218261243504995943, −12.3580133855055983549301951355, −11.95630308410066733869851386254, −11.10273963896022650274623829605, −10.383913405578944031359363840809, −9.57430330549393513565552214228, −8.213189040099379020061657362699, −7.82345006110453997450546053889, −6.871160193001640260936337854410, −6.24110571257073693379904775804, −4.45120343143096599500228323797, −3.7902583770477985014746838318, −2.716074929000348068051794805269, −1.85389781418313377877582499505, −0.79600468434948862841233428133, 0.140859215209250818357769725008, 1.41579968312798574247016094443, 2.9821692406555935760009684033, 4.283402506504852267565629792365, 4.888455701822619732439066045244, 5.60633551780168964754315424848, 6.4811298213357506716628060752, 7.82147242603519047395152218665, 8.42986917001784877090977136748, 9.38822053195862155058397720559, 9.55445791560425960838552419834, 10.77282700725131545891150861990, 11.86834053464365981674995011885, 12.3892783362808899430842221276, 13.76502843836046507999343733700, 14.70954069539950389774621996865, 15.17848490944776527439171650583, 15.90610499124568293733124712779, 16.594705711491229095258862710573, 17.21823512436465748700855103890, 17.90895242267070942212239536073, 19.1661889925836135323736103297, 19.68476335851704623495595693876, 20.45506783615211594345093112499, 21.582848913943953100071878991366

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