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Properties

Label 1-1033-1033.127-r0-0-0
Degree $1$
Conductor $1033$
Sign $0.666 + 0.745i$
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  + (0.601 + 0.798i)2-s + (0.0851 + 0.996i)3-s + (−0.276 + 0.961i)4-s + (0.702 − 0.711i)5-s + (−0.744 + 0.667i)6-s + (−0.391 − 0.920i)7-s + (−0.934 + 0.357i)8-s + (−0.985 + 0.169i)9-s + (0.991 + 0.133i)10-s + (0.468 − 0.883i)11-s + (−0.981 − 0.193i)12-s + (0.169 + 0.985i)13-s + (0.5 − 0.866i)14-s + (0.768 + 0.639i)15-s + (−0.847 − 0.531i)16-s + (0.413 − 0.910i)17-s + ⋯
L(s)  = 1  + (0.601 + 0.798i)2-s + (0.0851 + 0.996i)3-s + (−0.276 + 0.961i)4-s + (0.702 − 0.711i)5-s + (−0.744 + 0.667i)6-s + (−0.391 − 0.920i)7-s + (−0.934 + 0.357i)8-s + (−0.985 + 0.169i)9-s + (0.991 + 0.133i)10-s + (0.468 − 0.883i)11-s + (−0.981 − 0.193i)12-s + (0.169 + 0.985i)13-s + (0.5 − 0.866i)14-s + (0.768 + 0.639i)15-s + (−0.847 − 0.531i)16-s + (0.413 − 0.910i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.666 + 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.666 + 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1033\)
Sign: $0.666 + 0.745i$
Analytic conductor: \(4.79723\)
Root analytic conductor: \(4.79723\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1033} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1033,\ (0:\ ),\ 0.666 + 0.745i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.976127494 + 0.8845106502i\)
\(L(\frac12)\) \(\approx\) \(1.976127494 + 0.8845106502i\)
\(L(1)\) \(\approx\) \(1.369221371 + 0.6853489507i\)
\(L(1)\) \(\approx\) \(1.369221371 + 0.6853489507i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1033 \( 1 \)
good2 \( 1 + (0.601 + 0.798i)T \)
3 \( 1 + (0.0851 + 0.996i)T \)
5 \( 1 + (0.702 - 0.711i)T \)
7 \( 1 + (-0.391 - 0.920i)T \)
11 \( 1 + (0.468 - 0.883i)T \)
13 \( 1 + (0.169 + 0.985i)T \)
17 \( 1 + (0.413 - 0.910i)T \)
19 \( 1 + (0.478 - 0.877i)T \)
23 \( 1 + (0.311 - 0.950i)T \)
29 \( 1 + (0.929 + 0.368i)T \)
31 \( 1 + (0.964 - 0.264i)T \)
37 \( 1 + (-0.0365 + 0.999i)T \)
41 \( 1 + (-0.345 - 0.938i)T \)
43 \( 1 + (-0.435 + 0.900i)T \)
47 \( 1 + (0.987 + 0.157i)T \)
53 \( 1 + (-0.967 - 0.252i)T \)
59 \( 1 + (-0.929 + 0.368i)T \)
61 \( 1 + (-0.989 + 0.145i)T \)
67 \( 1 + (0.685 - 0.728i)T \)
71 \( 1 + (-0.702 - 0.711i)T \)
73 \( 1 + (0.999 - 0.0365i)T \)
79 \( 1 + (0.531 + 0.847i)T \)
83 \( 1 + (-0.157 - 0.987i)T \)
89 \( 1 + (0.983 + 0.181i)T \)
97 \( 1 + (0.648 + 0.760i)T \)
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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

−21.590328330257979337758473888627, −20.69132276779091834065695837476, −19.81567742537130558024405359326, −19.150320845089692832249546642422, −18.51798212601618273430254154498, −17.826222988111289800156081639774, −17.234016007882978533902852432263, −15.5155516309445915590192882812, −14.95414310111012539347251268055, −14.15917000488521593084708957543, −13.48753255718154912356725884916, −12.54165360999617778073878705049, −12.257448181943845089196742564838, −11.29586664841536219767330465937, −10.25308451098224134435050093841, −9.65971271929847890742642417752, −8.65630510039519801066128808071, −7.53178363621873846408243723208, −6.38799948022112999978672764170, −5.95258479181776442633173469646, −5.168031867950975429669020004299, −3.54132414571721856158832186554, −2.89620862397196647008286878312, −2.01238312717002916395704767461, −1.28501070492634448079346125027, 0.80500465520722963976659370343, 2.72624799803673313307812661753, 3.51178421262374364833357852801, 4.64878837218181363117811629802, 4.847892678273249852246908502763, 6.15284716111447263073789950846, 6.651301612689423240735975527325, 7.94596882033785341525107471829, 8.942580891694424109006621870680, 9.311697365232061770421736169670, 10.33912851685669259184943447798, 11.412352838392690876483622219862, 12.20333716131969500032650078995, 13.479204810581086486513318520259, 13.87256992912009896200404581231, 14.33988264271159063350508991876, 15.663986058848437114540743511238, 16.1775903483188475842737240854, 16.91430999280242058708905256755, 17.1023574534219046313849467094, 18.347747455337354725832838633801, 19.590684475911228338756625973429, 20.51033231462468217702428978086, 21.00629493560274546507604114818, 21.76510423060051522085129586936

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