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Properties

Label 1-1033-1033.1016-r0-0-0
Degree $1$
Conductor $1033$
Sign $0.718 + 0.695i$
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  + (−0.368 + 0.929i)2-s + (0.950 + 0.311i)3-s + (−0.728 − 0.685i)4-s + (0.910 − 0.413i)5-s + (−0.639 + 0.768i)6-s + (−0.989 − 0.145i)7-s + (0.905 − 0.424i)8-s + (0.806 + 0.591i)9-s + (0.0486 + 0.998i)10-s + (−0.446 − 0.894i)11-s + (−0.478 − 0.877i)12-s + (−0.591 + 0.806i)13-s + (0.5 − 0.866i)14-s + (0.994 − 0.109i)15-s + (0.0608 + 0.998i)16-s + (−0.0121 − 0.999i)17-s + ⋯
L(s)  = 1  + (−0.368 + 0.929i)2-s + (0.950 + 0.311i)3-s + (−0.728 − 0.685i)4-s + (0.910 − 0.413i)5-s + (−0.639 + 0.768i)6-s + (−0.989 − 0.145i)7-s + (0.905 − 0.424i)8-s + (0.806 + 0.591i)9-s + (0.0486 + 0.998i)10-s + (−0.446 − 0.894i)11-s + (−0.478 − 0.877i)12-s + (−0.591 + 0.806i)13-s + (0.5 − 0.866i)14-s + (0.994 − 0.109i)15-s + (0.0608 + 0.998i)16-s + (−0.0121 − 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1033\)
Sign: $0.718 + 0.695i$
Analytic conductor: \(4.79723\)
Root analytic conductor: \(4.79723\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1033} (1016, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1033,\ (0:\ ),\ 0.718 + 0.695i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.606479883 + 0.6495312297i\)
\(L(\frac12)\) \(\approx\) \(1.606479883 + 0.6495312297i\)
\(L(1)\) \(\approx\) \(1.153015544 + 0.4304820196i\)
\(L(1)\) \(\approx\) \(1.153015544 + 0.4304820196i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1033 \( 1 \)
good2 \( 1 + (-0.368 + 0.929i)T \)
3 \( 1 + (0.950 + 0.311i)T \)
5 \( 1 + (0.910 - 0.413i)T \)
7 \( 1 + (-0.989 - 0.145i)T \)
11 \( 1 + (-0.446 - 0.894i)T \)
13 \( 1 + (-0.591 + 0.806i)T \)
17 \( 1 + (-0.0121 - 0.999i)T \)
19 \( 1 + (0.925 + 0.379i)T \)
23 \( 1 + (-0.169 + 0.985i)T \)
29 \( 1 + (0.961 - 0.276i)T \)
31 \( 1 + (-0.995 - 0.0972i)T \)
37 \( 1 + (0.833 - 0.551i)T \)
41 \( 1 + (0.915 + 0.402i)T \)
43 \( 1 + (0.992 + 0.121i)T \)
47 \( 1 + (0.996 + 0.0851i)T \)
53 \( 1 + (-0.813 - 0.581i)T \)
59 \( 1 + (-0.961 - 0.276i)T \)
61 \( 1 + (0.694 - 0.719i)T \)
67 \( 1 + (0.531 + 0.847i)T \)
71 \( 1 + (-0.910 - 0.413i)T \)
73 \( 1 + (0.551 - 0.833i)T \)
79 \( 1 + (0.998 + 0.0608i)T \)
83 \( 1 + (0.0851 + 0.996i)T \)
89 \( 1 + (0.217 - 0.976i)T \)
97 \( 1 + (-0.736 + 0.676i)T \)
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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

−21.39503767544669903434008761884, −20.40195219102257407796943890916, −20.03870871132158249051638717924, −19.21736994884642439047512377607, −18.4443413231054857960004207698, −17.89326727736942952847230275813, −17.1544255662296612127583379928, −15.95854618160212180176200034432, −14.98106931723096152591964602536, −14.18674789299814552334591244663, −13.37639137509496016515442556715, −12.58961819399059780384536014230, −12.43231342359095859952345861034, −10.69717471503965735754646875065, −10.14759950063294815180374021499, −9.52321694727402989913578054183, −8.86644622307806542460289808968, −7.747683529478726927581943309534, −7.07460211260876212150148047466, −5.940259044885373735011902412013, −4.68866691568637059875862346549, −3.54345750932688700687088014703, −2.6511359658171618899514995943, −2.32482376067134131757906946303, −1.049673797187099277316092348352, 0.913755844133175538435885945239, 2.23764131592386434069757289926, 3.266772439084222970325099370257, 4.39690418570499026501905639759, 5.34628987545505836566212911558, 6.11974353934940520736544616479, 7.17006689028916512213025496206, 7.84077073705427983951291987518, 8.96877773975673353865619312732, 9.50428103998651644673230605476, 9.82631082115716561751817177857, 10.945050112800983257329336059973, 12.53295513816496423407673913251, 13.400831171400220227358773529484, 13.93756170792309721285217586808, 14.370994940084892074465401262301, 15.687547805035155146663952158295, 16.18737649949113724550895859223, 16.627899238512846079122488108854, 17.75915150647123253136387172553, 18.55711057011212443620783594230, 19.26579843679593682956694020659, 19.9409418799274670208126108956, 20.87428303226543559030232495074, 21.80391602383939529614566675525

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