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Properties

Label 1-1033-1033.111-r0-0-0
Degree $1$
Conductor $1033$
Sign $-0.561 - 0.827i$
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  + (0.435 − 0.900i)2-s + (−0.776 − 0.630i)3-s + (−0.620 − 0.783i)4-s + (−0.0486 + 0.998i)5-s + (−0.905 + 0.424i)6-s + (0.997 + 0.0729i)7-s + (−0.976 + 0.217i)8-s + (0.205 + 0.978i)9-s + (0.877 + 0.478i)10-s + (−0.685 − 0.728i)11-s + (−0.0121 + 0.999i)12-s + (−0.978 + 0.205i)13-s + (0.5 − 0.866i)14-s + (0.667 − 0.744i)15-s + (−0.229 + 0.973i)16-s + (0.964 − 0.264i)17-s + ⋯
L(s)  = 1  + (0.435 − 0.900i)2-s + (−0.776 − 0.630i)3-s + (−0.620 − 0.783i)4-s + (−0.0486 + 0.998i)5-s + (−0.905 + 0.424i)6-s + (0.997 + 0.0729i)7-s + (−0.976 + 0.217i)8-s + (0.205 + 0.978i)9-s + (0.877 + 0.478i)10-s + (−0.685 − 0.728i)11-s + (−0.0121 + 0.999i)12-s + (−0.978 + 0.205i)13-s + (0.5 − 0.866i)14-s + (0.667 − 0.744i)15-s + (−0.229 + 0.973i)16-s + (0.964 − 0.264i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.561 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.561 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1033\)
Sign: $-0.561 - 0.827i$
Analytic conductor: \(4.79723\)
Root analytic conductor: \(4.79723\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1033} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1033,\ (0:\ ),\ -0.561 - 0.827i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5333715121 - 1.005956752i\)
\(L(\frac12)\) \(\approx\) \(0.5333715121 - 1.005956752i\)
\(L(1)\) \(\approx\) \(0.7800591302 - 0.5475943585i\)
\(L(1)\) \(\approx\) \(0.7800591302 - 0.5475943585i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1033 \( 1 \)
good2 \( 1 + (0.435 - 0.900i)T \)
3 \( 1 + (-0.776 - 0.630i)T \)
5 \( 1 + (-0.0486 + 0.998i)T \)
7 \( 1 + (0.997 + 0.0729i)T \)
11 \( 1 + (-0.685 - 0.728i)T \)
13 \( 1 + (-0.978 + 0.205i)T \)
17 \( 1 + (0.964 - 0.264i)T \)
19 \( 1 + (-0.658 + 0.752i)T \)
23 \( 1 + (-0.571 + 0.820i)T \)
29 \( 1 + (0.121 - 0.992i)T \)
31 \( 1 + (0.541 - 0.840i)T \)
37 \( 1 + (0.957 - 0.288i)T \)
41 \( 1 + (0.950 - 0.311i)T \)
43 \( 1 + (0.894 - 0.446i)T \)
47 \( 1 + (-0.954 - 0.299i)T \)
53 \( 1 + (0.889 - 0.457i)T \)
59 \( 1 + (-0.121 - 0.992i)T \)
61 \( 1 + (-0.391 - 0.920i)T \)
67 \( 1 + (0.241 - 0.970i)T \)
71 \( 1 + (0.0486 + 0.998i)T \)
73 \( 1 + (0.288 - 0.957i)T \)
79 \( 1 + (0.973 - 0.229i)T \)
83 \( 1 + (-0.299 - 0.954i)T \)
89 \( 1 + (0.994 + 0.109i)T \)
97 \( 1 + (0.591 + 0.806i)T \)
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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

−21.77572764236563610446657278344, −21.23758638517681763205184487020, −20.69509324101117431065565879190, −19.69122518253268793751878309473, −18.05240935873925349824192037626, −17.812988084429218821581905484568, −16.90339695661059261420643871521, −16.48566852172349202724993020066, −15.58139613201471584431135244717, −14.8887084494767766469979323950, −14.27655964456683927904569287155, −12.93032897767024269321777879972, −12.42970515711234403538201579762, −11.78824191692767862356405528750, −10.5850991670292562059716944551, −9.7357154697385807144141568816, −8.8008836212508238258705726305, −7.96997191740381985820977563190, −7.21209311963090378092490496581, −6.02026086000156861688826989930, −5.17929293398071424876815212015, −4.6989247586268380938336682582, −4.1767256465141798472224503252, −2.63753153985959232350962080758, −0.96615935052222561274805014337, 0.57774873121515123318406449296, 1.974701634540139666357652227510, 2.47305158731283984062846531579, 3.73877662794986899429303991901, 4.78833429936032773868483245726, 5.66840230934482384167988938907, 6.222980332057139301605721515282, 7.6434819554023419903636570777, 8.00679257787753749413500699805, 9.66886330223920078918913607413, 10.39198341090312155126758196027, 11.117523462595284632015330795487, 11.6909661859758714158286320325, 12.30733426849159214293788761975, 13.3631924449414524052481724585, 14.10131428438638191236633766065, 14.65834668518859598968070730565, 15.64014895626131505702661406886, 16.900098489273453259640008869005, 17.7022027204692807491858423862, 18.34694649774742139726323391676, 19.00804088124602152548759189793, 19.427224559582903496673152245088, 20.76616072059244516295462851493, 21.511390802110642155883438103804

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