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Properties

Label 1-1033-1033.145-r0-0-0
Degree $1$
Conductor $1033$
Sign $0.648 - 0.761i$
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  + (0.995 − 0.0972i)2-s + (−0.883 + 0.468i)3-s + (0.981 − 0.193i)4-s + (0.402 − 0.915i)5-s + (−0.833 + 0.551i)6-s + (0.581 − 0.813i)7-s + (0.957 − 0.288i)8-s + (0.561 − 0.827i)9-s + (0.311 − 0.950i)10-s + (0.991 − 0.133i)11-s + (−0.776 + 0.630i)12-s + (0.827 + 0.561i)13-s + (0.5 − 0.866i)14-s + (0.0729 + 0.997i)15-s + (0.925 − 0.379i)16-s + (0.760 − 0.648i)17-s + ⋯
L(s)  = 1  + (0.995 − 0.0972i)2-s + (−0.883 + 0.468i)3-s + (0.981 − 0.193i)4-s + (0.402 − 0.915i)5-s + (−0.833 + 0.551i)6-s + (0.581 − 0.813i)7-s + (0.957 − 0.288i)8-s + (0.561 − 0.827i)9-s + (0.311 − 0.950i)10-s + (0.991 − 0.133i)11-s + (−0.776 + 0.630i)12-s + (0.827 + 0.561i)13-s + (0.5 − 0.866i)14-s + (0.0729 + 0.997i)15-s + (0.925 − 0.379i)16-s + (0.760 − 0.648i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.648 - 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.648 - 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1033\)
Sign: $0.648 - 0.761i$
Analytic conductor: \(4.79723\)
Root analytic conductor: \(4.79723\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1033} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1033,\ (0:\ ),\ 0.648 - 0.761i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.610588546 - 1.205885010i\)
\(L(\frac12)\) \(\approx\) \(2.610588546 - 1.205885010i\)
\(L(1)\) \(\approx\) \(1.843664716 - 0.4140625178i\)
\(L(1)\) \(\approx\) \(1.843664716 - 0.4140625178i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1033 \( 1 \)
good2 \( 1 + (0.995 - 0.0972i)T \)
3 \( 1 + (-0.883 + 0.468i)T \)
5 \( 1 + (0.402 - 0.915i)T \)
7 \( 1 + (0.581 - 0.813i)T \)
11 \( 1 + (0.991 - 0.133i)T \)
13 \( 1 + (0.827 + 0.561i)T \)
17 \( 1 + (0.760 - 0.648i)T \)
19 \( 1 + (-0.820 - 0.571i)T \)
23 \( 1 + (-0.446 + 0.894i)T \)
29 \( 1 + (0.510 + 0.859i)T \)
31 \( 1 + (-0.806 + 0.591i)T \)
37 \( 1 + (-0.791 - 0.611i)T \)
41 \( 1 + (-0.435 + 0.900i)T \)
43 \( 1 + (-0.711 - 0.702i)T \)
47 \( 1 + (0.973 + 0.229i)T \)
53 \( 1 + (-0.994 + 0.109i)T \)
59 \( 1 + (-0.510 + 0.859i)T \)
61 \( 1 + (0.872 + 0.489i)T \)
67 \( 1 + (-0.877 - 0.478i)T \)
71 \( 1 + (-0.402 - 0.915i)T \)
73 \( 1 + (0.611 + 0.791i)T \)
79 \( 1 + (-0.379 + 0.925i)T \)
83 \( 1 + (0.229 + 0.973i)T \)
89 \( 1 + (-0.145 + 0.989i)T \)
97 \( 1 + (0.783 - 0.620i)T \)
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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

−21.94297393043122096906512239776, −21.23779797822853194992787269517, −20.39542384909972762959106151237, −18.9878463447014768794205498996, −18.76450294557137862244910672921, −17.569379952050980604592269475441, −17.12405174978161942732356942358, −16.100379545450665453389712382557, −15.170105789308828611382736984502, −14.57276482432201310189855083536, −13.83480686857316899552735364636, −12.85959166142608264413794524419, −12.1524147097709170566403676790, −11.52131471696901631900206234851, −10.733704041193450561709645708, −10.10549398479050765854343082693, −8.4452702681017095996602590569, −7.641483217416959177729487563138, −6.49578791782390035593782488096, −6.13407075233760268587226996932, −5.47957796840726738616126295866, −4.34727798912475410996181073487, −3.38890281887392189526012032695, −2.12886053634168940046177673817, −1.55393353492989180230526205616, 1.11218942224422806046960666550, 1.64443790453339440311817212178, 3.51489205893965226433446494837, 4.16724231327952651648222610602, 4.90668073671685683465487534948, 5.611471676976212061115416238171, 6.514970882723646127020118871230, 7.24853258233878993889677262543, 8.638860882861064031183018555781, 9.58208556456141538648622856153, 10.54330144197377537502690142070, 11.246105973873753136855541939421, 11.94565022634345306928765409412, 12.62504284202209338405957454910, 13.695104083660808670183137187868, 14.11836844330637180477163516471, 15.173407819236229869445534343185, 16.17654520580884655884030742322, 16.59534462856303076535780372184, 17.24556324287866631891161401968, 18.10444709759724019515690174900, 19.4725089951359281669156212989, 20.24358970629854675779827338792, 20.93857240109749496857364965520, 21.49769567548596899588046188786

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