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Properties

Label 1-1021-1021.100-r0-0-0
Degree $1$
Conductor $1021$
Sign $0.999 + 0.0424i$
Analytic cond. $4.74150$
Root an. cond. $4.74150$
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  + (−0.572 − 0.819i)2-s + (−0.982 − 0.183i)3-s + (−0.343 + 0.938i)4-s + (−0.582 + 0.812i)5-s + (0.412 + 0.911i)6-s + (−0.989 + 0.147i)7-s + (0.966 − 0.255i)8-s + (0.932 + 0.361i)9-s + (0.999 + 0.0123i)10-s + (0.980 + 0.195i)11-s + (0.510 − 0.859i)12-s + (0.602 + 0.798i)13-s + (0.687 + 0.726i)14-s + (0.722 − 0.691i)15-s + (−0.763 − 0.645i)16-s + (0.401 − 0.916i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.819i)2-s + (−0.982 − 0.183i)3-s + (−0.343 + 0.938i)4-s + (−0.582 + 0.812i)5-s + (0.412 + 0.911i)6-s + (−0.989 + 0.147i)7-s + (0.966 − 0.255i)8-s + (0.932 + 0.361i)9-s + (0.999 + 0.0123i)10-s + (0.980 + 0.195i)11-s + (0.510 − 0.859i)12-s + (0.602 + 0.798i)13-s + (0.687 + 0.726i)14-s + (0.722 − 0.691i)15-s + (−0.763 − 0.645i)16-s + (0.401 − 0.916i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0424i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0424i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $0.999 + 0.0424i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ 0.999 + 0.0424i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6089860146 + 0.01293287765i\)
\(L(\frac12)\) \(\approx\) \(0.6089860146 + 0.01293287765i\)
\(L(1)\) \(\approx\) \(0.5404121417 - 0.07861473002i\)
\(L(1)\) \(\approx\) \(0.5404121417 - 0.07861473002i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (-0.572 - 0.819i)T \)
3 \( 1 + (-0.982 - 0.183i)T \)
5 \( 1 + (-0.582 + 0.812i)T \)
7 \( 1 + (-0.989 + 0.147i)T \)
11 \( 1 + (0.980 + 0.195i)T \)
13 \( 1 + (0.602 + 0.798i)T \)
17 \( 1 + (0.401 - 0.916i)T \)
19 \( 1 + (0.779 + 0.626i)T \)
23 \( 1 + (-0.177 - 0.984i)T \)
29 \( 1 + (-0.622 + 0.782i)T \)
31 \( 1 + (0.836 + 0.547i)T \)
37 \( 1 + (0.823 + 0.567i)T \)
41 \( 1 + (-0.779 - 0.626i)T \)
43 \( 1 + (-0.963 - 0.267i)T \)
47 \( 1 + (-0.972 + 0.231i)T \)
53 \( 1 + (0.990 + 0.135i)T \)
59 \( 1 + (0.794 - 0.607i)T \)
61 \( 1 + (0.285 - 0.958i)T \)
67 \( 1 + (-0.104 - 0.994i)T \)
71 \( 1 + (-0.966 + 0.255i)T \)
73 \( 1 + (0.687 - 0.726i)T \)
79 \( 1 + (0.975 + 0.219i)T \)
83 \( 1 + (0.650 - 0.759i)T \)
89 \( 1 + (-0.696 - 0.717i)T \)
97 \( 1 + (-0.552 + 0.833i)T \)
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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

−21.90874481462061625284445615778, −20.70208360524068074700753803721, −19.63022310188336195400492787124, −19.38112272143896408985750137032, −18.232411419872647152485383680825, −17.48787321916559376440518744416, −16.65284109697865480174102447656, −16.42857486454540871349309149357, −15.46510461852765888876293543025, −15.06825261454156033479453486693, −13.43743586616875778410450139005, −13.06079528569396589625726149721, −11.837909755041868964711121260431, −11.26689940230150989519782285724, −10.04540811230867123756190385403, −9.60758479092161683770515364332, −8.63180179993592778260363013723, −7.71906589059875220402893524570, −6.825728138993297348792053512871, −5.96256629593514883683564402158, −5.44999901918843546392881611941, −4.26592119272009882021767486443, −3.57760026726998102195539914504, −1.34315946135050049974756664671, −0.63016988959078002138010261204, 0.76516048183775079133501699134, 1.92982714034405327831202084056, 3.22714992321792347515600644060, 3.83114622410775587558254608063, 4.903834718205077232435520864141, 6.42813481585730806523688905146, 6.78572920919833990514909737606, 7.710083336058595108915394312356, 8.89951945051516906288554039588, 9.83339601416187366143327350976, 10.36494529683888340832192039600, 11.43564936829169683965555154010, 11.816016400247760024669973174686, 12.43683444170269150713905450859, 13.494103430404458152181776364838, 14.33494050665457416429288310551, 15.691798592452884464201470042778, 16.445121284338176541330422229479, 16.792679374883105562088496005005, 18.12115870146775064210205962221, 18.4827157812411559832458548898, 19.07878902143931203497253845836, 19.8590675087414312283013466744, 20.75134677269257266340556277255, 21.86951726242884005189499521996

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