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Properties

Label 1-1033-1033.70-r1-0-0
Degree $1$
Conductor $1033$
Sign $0.962 - 0.272i$
Analytic cond. $111.011$
Root an. cond. $111.011$
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  + (0.992 + 0.121i)2-s + (0.571 − 0.820i)3-s + (0.970 + 0.241i)4-s + (−0.869 + 0.494i)5-s + (0.667 − 0.744i)6-s + (−0.920 − 0.391i)7-s + (0.934 + 0.357i)8-s + (−0.345 − 0.938i)9-s + (−0.922 + 0.385i)10-s + (0.974 − 0.223i)11-s + (0.752 − 0.658i)12-s + (0.908 + 0.418i)13-s + (−0.866 − 0.5i)14-s + (−0.0911 + 0.995i)15-s + (0.883 + 0.468i)16-s + (−0.0972 + 0.995i)17-s + ⋯
L(s)  = 1  + (0.992 + 0.121i)2-s + (0.571 − 0.820i)3-s + (0.970 + 0.241i)4-s + (−0.869 + 0.494i)5-s + (0.667 − 0.744i)6-s + (−0.920 − 0.391i)7-s + (0.934 + 0.357i)8-s + (−0.345 − 0.938i)9-s + (−0.922 + 0.385i)10-s + (0.974 − 0.223i)11-s + (0.752 − 0.658i)12-s + (0.908 + 0.418i)13-s + (−0.866 − 0.5i)14-s + (−0.0911 + 0.995i)15-s + (0.883 + 0.468i)16-s + (−0.0972 + 0.995i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.962 - 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.962 - 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1033\)
Sign: $0.962 - 0.272i$
Analytic conductor: \(111.011\)
Root analytic conductor: \(111.011\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1033} (70, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1033,\ (1:\ ),\ 0.962 - 0.272i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.633130238 - 0.6437146622i\)
\(L(\frac12)\) \(\approx\) \(4.633130238 - 0.6437146622i\)
\(L(1)\) \(\approx\) \(2.130802000 - 0.2340896294i\)
\(L(1)\) \(\approx\) \(2.130802000 - 0.2340896294i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1033 \( 1 \)
good2 \( 1 + (0.992 + 0.121i)T \)
3 \( 1 + (0.571 - 0.820i)T \)
5 \( 1 + (-0.869 + 0.494i)T \)
7 \( 1 + (-0.920 - 0.391i)T \)
11 \( 1 + (0.974 - 0.223i)T \)
13 \( 1 + (0.908 + 0.418i)T \)
17 \( 1 + (-0.0972 + 0.995i)T \)
19 \( 1 + (-0.999 - 0.0243i)T \)
23 \( 1 + (0.837 - 0.546i)T \)
29 \( 1 + (0.115 + 0.993i)T \)
31 \( 1 + (0.702 + 0.711i)T \)
37 \( 1 + (0.0365 + 0.999i)T \)
41 \( 1 + (-0.169 - 0.985i)T \)
43 \( 1 + (-0.827 - 0.561i)T \)
47 \( 1 + (0.994 + 0.103i)T \)
53 \( 1 + (0.862 + 0.505i)T \)
59 \( 1 + (-0.993 - 0.115i)T \)
61 \( 1 + (0.145 - 0.989i)T \)
67 \( 1 + (0.526 - 0.850i)T \)
71 \( 1 + (0.494 - 0.869i)T \)
73 \( 1 + (0.732 - 0.680i)T \)
79 \( 1 + (-0.293 + 0.955i)T \)
83 \( 1 + (0.630 - 0.776i)T \)
89 \( 1 + (0.823 + 0.566i)T \)
97 \( 1 + (-0.429 + 0.902i)T \)
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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

−21.334587568571750422074665478541, −20.74832223920549893527454064700, −19.909541737805953595102507111969, −19.512030412260055983993846672292, −18.75442957856857184254654070609, −16.99002569408334302428663624963, −16.43156574598944939118347282595, −15.51131578690685391292489492990, −15.37677866287924625636292361226, −14.419253044800609478884990900127, −13.37089116285635592580469558016, −12.92349213874257198252950244709, −11.76775512963319168167870546906, −11.330968115699363914752337148018, −10.23012657856196594645815130814, −9.334650835737350395980321684673, −8.580918697095502351451194494733, −7.543729901831055964942712346405, −6.528696243708779951490946605669, −5.5953351087267280078750899712, −4.57050122181044517446104733524, −3.92303418003548351836219068647, −3.259184275846633310472781214084, −2.34087104665388344433932915286, −0.82233874417160380576314835546, 0.86552749179590725959086476789, 1.99842258840050756181773910865, 3.27835300421178923581212592508, 3.56235788748373995928913785341, 4.46380606212777803011098276822, 6.200016958232733259094233284432, 6.584892963760302996553364545792, 7.12203941571241990806966695560, 8.29554163545068451926668432681, 8.88118400867632090444665647666, 10.48460098988517595753046978271, 11.10959752120607252598070131746, 12.22937564656346702672808087589, 12.52981808040537725390042498429, 13.58227455309353146918344222891, 14.070426746887001867452486987807, 15.01550281829582916591661151271, 15.47622802771858492462291722385, 16.59535563656068365793047242904, 17.16493052151988193102724716469, 18.663967461112677726268542389403, 19.174089542347567841810988598824, 19.79368196233089636546433444307, 20.38892281276179214381287883935, 21.46521436156919668500888229838

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