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Properties

Label 1-1021-1021.767-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: $\chi_{1021} (767, \cdot )$
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

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

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