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Properties

Label 1-1021-1021.205-r0-0-0
Degree $1$
Conductor $1021$
Sign $0.964 - 0.265i$
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  + (0.412 − 0.911i)2-s + (0.445 + 0.895i)3-s + (−0.659 − 0.751i)4-s + (0.531 + 0.846i)5-s + (0.999 − 0.0369i)6-s + (0.0554 − 0.998i)7-s + (−0.956 + 0.291i)8-s + (−0.602 + 0.798i)9-s + (0.990 − 0.135i)10-s + (−0.562 − 0.826i)11-s + (0.378 − 0.925i)12-s + (−0.739 + 0.673i)13-s + (−0.886 − 0.462i)14-s + (−0.521 + 0.853i)15-s + (−0.128 + 0.991i)16-s + (0.985 + 0.171i)17-s + ⋯
L(s)  = 1  + (0.412 − 0.911i)2-s + (0.445 + 0.895i)3-s + (−0.659 − 0.751i)4-s + (0.531 + 0.846i)5-s + (0.999 − 0.0369i)6-s + (0.0554 − 0.998i)7-s + (−0.956 + 0.291i)8-s + (−0.602 + 0.798i)9-s + (0.990 − 0.135i)10-s + (−0.562 − 0.826i)11-s + (0.378 − 0.925i)12-s + (−0.739 + 0.673i)13-s + (−0.886 − 0.462i)14-s + (−0.521 + 0.853i)15-s + (−0.128 + 0.991i)16-s + (0.985 + 0.171i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $0.964 - 0.265i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ 0.964 - 0.265i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.091008800 - 0.2825659206i\)
\(L(\frac12)\) \(\approx\) \(2.091008800 - 0.2825659206i\)
\(L(1)\) \(\approx\) \(1.463476048 - 0.2448866804i\)
\(L(1)\) \(\approx\) \(1.463476048 - 0.2448866804i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (0.412 - 0.911i)T \)
3 \( 1 + (0.445 + 0.895i)T \)
5 \( 1 + (0.531 + 0.846i)T \)
7 \( 1 + (0.0554 - 0.998i)T \)
11 \( 1 + (-0.562 - 0.826i)T \)
13 \( 1 + (-0.739 + 0.673i)T \)
17 \( 1 + (0.985 + 0.171i)T \)
19 \( 1 + (0.389 - 0.920i)T \)
23 \( 1 + (0.923 + 0.384i)T \)
29 \( 1 + (0.892 + 0.451i)T \)
31 \( 1 + (0.996 - 0.0861i)T \)
37 \( 1 + (0.936 - 0.349i)T \)
41 \( 1 + (-0.389 + 0.920i)T \)
43 \( 1 + (0.987 + 0.159i)T \)
47 \( 1 + (0.843 - 0.536i)T \)
53 \( 1 + (0.0799 - 0.996i)T \)
59 \( 1 + (0.622 + 0.782i)T \)
61 \( 1 + (0.0431 + 0.999i)T \)
67 \( 1 + (0.913 - 0.406i)T \)
71 \( 1 + (0.956 - 0.291i)T \)
73 \( 1 + (-0.886 + 0.462i)T \)
79 \( 1 + (-0.763 - 0.645i)T \)
83 \( 1 + (-0.998 - 0.0615i)T \)
89 \( 1 + (0.816 + 0.577i)T \)
97 \( 1 + (0.153 + 0.988i)T \)
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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

−21.62685087270269794854359373327, −20.847933543206849155429513712453, −20.33026826279803969549073347995, −18.9675622523486131803040172961, −18.4698663057874319369925818248, −17.4597461405148157759890643242, −17.22618699099098785408640869127, −15.98344393420378363163379018607, −15.2794615052851232404699743771, −14.47415957701258349784627730321, −13.83146956574460983113951064544, −12.75928080492741343423311194954, −12.48884640630127219603890115722, −11.93154182795071060498361900190, −9.95620276839515581050234430713, −9.3413989898674059639557269649, −8.324520590869950829779345944290, −7.92371117270980590892767189964, −6.9800341486381594105369483176, −5.874022159501169294075280301051, −5.416028579537474405308242604256, −4.5090167662767852694121721182, −2.9937759858451281979438100704, −2.34643124183259376406654605209, −0.92923060964828976550259154682, 1.04973926420396869218944098179, 2.54639588479881127808959979894, 2.99901120418085790219865158764, 3.90346918342316863222568972976, 4.83690379858644309651589191292, 5.58769165145716822161824206834, 6.77888728470283650319233657010, 7.86554453736554528343590525680, 9.0260675582404544956392273435, 9.83549036410700430741568531156, 10.34002992161048726602019926863, 11.04023600208126230445376230673, 11.68035385945273414677212510519, 13.17166181648906961284498451428, 13.70113114199851950393268726854, 14.35607757324347084707396833911, 14.88915588720560074513339533744, 15.96354581329522834576936522989, 16.94213167536339276662904974788, 17.6991753135677253359104186546, 18.81384772229387382100170466926, 19.390944144960166682107119351706, 20.05123914845038837298188265859, 21.20085826663398683005581197658, 21.30432865952431976039454248004

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