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Properties

Label 1-1012-1012.543-r0-0-0
Degree $1$
Conductor $1012$
Sign $-0.759 + 0.650i$
Analytic cond. $4.69970$
Root an. cond. $4.69970$
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  + (−0.696 + 0.717i)3-s + (0.0285 − 0.999i)5-s + (0.974 + 0.226i)7-s + (−0.0285 − 0.999i)9-s + (−0.921 − 0.389i)13-s + (0.696 + 0.717i)15-s + (−0.993 + 0.113i)17-s + (−0.870 + 0.491i)19-s + (−0.841 + 0.540i)21-s + (−0.998 − 0.0570i)25-s + (0.736 + 0.676i)27-s + (−0.870 − 0.491i)29-s + (−0.897 + 0.441i)31-s + (0.254 − 0.967i)35-s + (−0.941 + 0.336i)37-s + ⋯
L(s)  = 1  + (−0.696 + 0.717i)3-s + (0.0285 − 0.999i)5-s + (0.974 + 0.226i)7-s + (−0.0285 − 0.999i)9-s + (−0.921 − 0.389i)13-s + (0.696 + 0.717i)15-s + (−0.993 + 0.113i)17-s + (−0.870 + 0.491i)19-s + (−0.841 + 0.540i)21-s + (−0.998 − 0.0570i)25-s + (0.736 + 0.676i)27-s + (−0.870 − 0.491i)29-s + (−0.897 + 0.441i)31-s + (0.254 − 0.967i)35-s + (−0.941 + 0.336i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.759 + 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.759 + 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1012\)    =    \(2^{2} \cdot 11 \cdot 23\)
Sign: $-0.759 + 0.650i$
Analytic conductor: \(4.69970\)
Root analytic conductor: \(4.69970\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1012} (543, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1012,\ (0:\ ),\ -0.759 + 0.650i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1395449075 + 0.3777994901i\)
\(L(\frac12)\) \(\approx\) \(0.1395449075 + 0.3777994901i\)
\(L(1)\) \(\approx\) \(0.7023204134 + 0.09944738568i\)
\(L(1)\) \(\approx\) \(0.7023204134 + 0.09944738568i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
23 \( 1 \)
good3 \( 1 + (-0.696 + 0.717i)T \)
5 \( 1 + (0.0285 - 0.999i)T \)
7 \( 1 + (0.974 + 0.226i)T \)
13 \( 1 + (-0.921 - 0.389i)T \)
17 \( 1 + (-0.993 + 0.113i)T \)
19 \( 1 + (-0.870 + 0.491i)T \)
29 \( 1 + (-0.870 - 0.491i)T \)
31 \( 1 + (-0.897 + 0.441i)T \)
37 \( 1 + (-0.941 + 0.336i)T \)
41 \( 1 + (0.941 + 0.336i)T \)
43 \( 1 + (-0.142 + 0.989i)T \)
47 \( 1 + (0.809 + 0.587i)T \)
53 \( 1 + (-0.516 + 0.856i)T \)
59 \( 1 + (-0.0855 + 0.996i)T \)
61 \( 1 + (0.985 + 0.170i)T \)
67 \( 1 + (0.841 - 0.540i)T \)
71 \( 1 + (0.254 + 0.967i)T \)
73 \( 1 + (0.198 + 0.980i)T \)
79 \( 1 + (-0.921 - 0.389i)T \)
83 \( 1 + (-0.564 + 0.825i)T \)
89 \( 1 + (0.142 - 0.989i)T \)
97 \( 1 + (0.564 + 0.825i)T \)
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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

−21.68655501483646462581202555707, −20.49147357043364767504934685600, −19.54139168849512788843617313769, −18.87748762232111108529879152506, −18.14277780253261062216266244160, −17.43277546469886967366252146283, −17.01969292399049529899855996358, −15.794659670216330384274493698119, −14.83999072111417010286569527383, −14.2417566468613154617930911407, −13.404176553173484244008511033262, −12.5076750604866945495313464019, −11.556258095789943501825138930117, −11.014138690784380244806200781811, −10.45215362431557059031925186046, −9.176284932928508307768723586275, −8.08877090741459615994259153095, −7.12396859028684033656294968550, −6.88390867526867699235743150897, −5.69265893739018555920482070761, −4.86617009569730942197997457859, −3.864424080619850885163918853558, −2.26026650314138469087727050444, −1.9573049536586659613741471878, −0.1849605064490978324713988707, 1.28283882195412947372588556151, 2.385436951876761492796289671888, 3.98198418768481876353273710344, 4.5803785940451218267160658586, 5.3186161150554199292147376571, 6.01894147103166831433220992229, 7.302808341293173347492798286854, 8.33285893005760348889755019582, 9.036992063442396259230082235543, 9.85843293146407215406092683745, 10.833963646867649444826651022347, 11.470023965586846197449648800843, 12.38726903502330687901044322139, 12.894005594941493655172570189802, 14.215282826312222455564300501260, 15.00291697611636141127749444068, 15.650889421022474373588908360860, 16.54242355367726299605711154047, 17.33992120711158063225809857758, 17.57668371599524896268757944589, 18.696051704836124563246006103231, 19.888463234531391705557725759029, 20.47128190353435122070766593062, 21.25999635553405501175556113995, 21.74816156838504511259880356335

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