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Properties

Label 1-1033-1033.114-r0-0-0
Degree $1$
Conductor $1033$
Sign $-0.300 - 0.953i$
Analytic cond. $4.79723$
Root an. cond. $4.79723$
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  + (−0.820 + 0.571i)2-s + (−0.995 − 0.0972i)3-s + (0.345 − 0.938i)4-s + (−0.973 − 0.229i)5-s + (0.872 − 0.489i)6-s + (−0.905 + 0.424i)7-s + (0.252 + 0.967i)8-s + (0.981 + 0.193i)9-s + (0.929 − 0.368i)10-s + (−0.334 + 0.942i)11-s + (−0.435 + 0.900i)12-s + (−0.193 + 0.981i)13-s + (0.5 − 0.866i)14-s + (0.946 + 0.322i)15-s + (−0.760 − 0.648i)16-s + (−0.883 + 0.468i)17-s + ⋯
L(s)  = 1  + (−0.820 + 0.571i)2-s + (−0.995 − 0.0972i)3-s + (0.345 − 0.938i)4-s + (−0.973 − 0.229i)5-s + (0.872 − 0.489i)6-s + (−0.905 + 0.424i)7-s + (0.252 + 0.967i)8-s + (0.981 + 0.193i)9-s + (0.929 − 0.368i)10-s + (−0.334 + 0.942i)11-s + (−0.435 + 0.900i)12-s + (−0.193 + 0.981i)13-s + (0.5 − 0.866i)14-s + (0.946 + 0.322i)15-s + (−0.760 − 0.648i)16-s + (−0.883 + 0.468i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1033\)
Sign: $-0.300 - 0.953i$
Analytic conductor: \(4.79723\)
Root analytic conductor: \(4.79723\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1033} (114, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1033,\ (0:\ ),\ -0.300 - 0.953i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06977663073 + 0.09509881036i\)
\(L(\frac12)\) \(\approx\) \(-0.06977663073 + 0.09509881036i\)
\(L(1)\) \(\approx\) \(0.2924959151 + 0.1555327417i\)
\(L(1)\) \(\approx\) \(0.2924959151 + 0.1555327417i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1033 \( 1 \)
good2 \( 1 + (-0.820 + 0.571i)T \)
3 \( 1 + (-0.995 - 0.0972i)T \)
5 \( 1 + (-0.973 - 0.229i)T \)
7 \( 1 + (-0.905 + 0.424i)T \)
11 \( 1 + (-0.334 + 0.942i)T \)
13 \( 1 + (-0.193 + 0.981i)T \)
17 \( 1 + (-0.883 + 0.468i)T \)
19 \( 1 + (-0.992 + 0.121i)T \)
23 \( 1 + (0.510 - 0.859i)T \)
29 \( 1 + (-0.978 + 0.205i)T \)
31 \( 1 + (0.728 + 0.685i)T \)
37 \( 1 + (-0.181 + 0.983i)T \)
41 \( 1 + (0.658 + 0.752i)T \)
43 \( 1 + (-0.157 + 0.987i)T \)
47 \( 1 + (-0.264 + 0.964i)T \)
53 \( 1 + (0.288 + 0.957i)T \)
59 \( 1 + (0.978 + 0.205i)T \)
61 \( 1 + (-0.744 + 0.667i)T \)
67 \( 1 + (0.402 + 0.915i)T \)
71 \( 1 + (0.973 - 0.229i)T \)
73 \( 1 + (-0.983 + 0.181i)T \)
79 \( 1 + (-0.648 - 0.760i)T \)
83 \( 1 + (0.964 - 0.264i)T \)
89 \( 1 + (-0.611 - 0.791i)T \)
97 \( 1 + (-0.991 - 0.133i)T \)
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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

−20.9575616021484471262930408060, −20.06526552058315150410085663888, −19.26746046784557735052125728206, −18.86392775468996032939654552081, −17.92526069528255651880461516855, −17.12926246115525465661628290267, −16.49414592433759777123606676310, −15.69443492795616924335424412658, −15.310053802596825573091898513229, −13.4155791492066510901607573234, −12.88630499729941863244008490436, −12.07305374733551680578735053409, −11.09949699725354521353159932132, −10.87065652704999684743899126067, −9.98813397941795810590779899101, −9.03931865351772889778389351840, −8.01300683048803425826975877919, −7.19125338658021022400684140894, −6.54581541921803820487807849310, −5.393068536887551144178261597881, −4.0483900420881765200368621177, −3.50614807451410600032372790843, −2.38947703284816701009025650256, −0.63746527387856943212869791229, −0.13652954333485067312056560594, 1.37336096295386666995423249635, 2.54053898256611039066711679060, 4.312985904320220394961249624484, 4.78451008853167210745077955174, 6.08303529074598654482569381117, 6.71124125291147674518438479805, 7.29664884469405517319536502112, 8.39900269989913207138413574758, 9.202747099145329902237136655112, 10.0934311846160232942543158004, 10.88735652769141720943626442506, 11.65739673172108809207826852602, 12.50462155674719918187910421155, 13.13130734545434201276176610059, 14.76242837384638073374326477545, 15.30779797663881578244990492648, 16.06622719235024583329851453237, 16.60802361572595910120775497028, 17.30270245577524889131003569141, 18.21583437164882598344479404463, 18.985754324256570300960320621733, 19.37650243087374486456710989670, 20.32449124440098614898242473734, 21.36252912433727217429492833568, 22.49460506141091315134374422238

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