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Properties

Label 1-1033-1033.144-r0-0-0
Degree $1$
Conductor $1033$
Sign $-0.993 - 0.110i$
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  + (−0.894 + 0.446i)2-s + (0.676 + 0.736i)3-s + (0.601 − 0.798i)4-s + (−0.925 + 0.379i)5-s + (−0.934 − 0.357i)6-s + (0.833 − 0.551i)7-s + (−0.181 + 0.983i)8-s + (−0.0851 + 0.996i)9-s + (0.658 − 0.752i)10-s + (−0.970 + 0.241i)11-s + (0.995 − 0.0972i)12-s + (0.0851 − 0.996i)13-s + (−0.5 + 0.866i)14-s + (−0.905 − 0.424i)15-s + (−0.276 − 0.961i)16-s + (−0.541 + 0.840i)17-s + ⋯
L(s)  = 1  + (−0.894 + 0.446i)2-s + (0.676 + 0.736i)3-s + (0.601 − 0.798i)4-s + (−0.925 + 0.379i)5-s + (−0.934 − 0.357i)6-s + (0.833 − 0.551i)7-s + (−0.181 + 0.983i)8-s + (−0.0851 + 0.996i)9-s + (0.658 − 0.752i)10-s + (−0.970 + 0.241i)11-s + (0.995 − 0.0972i)12-s + (0.0851 − 0.996i)13-s + (−0.5 + 0.866i)14-s + (−0.905 − 0.424i)15-s + (−0.276 − 0.961i)16-s + (−0.541 + 0.840i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1033\)
Sign: $-0.993 - 0.110i$
Analytic conductor: \(4.79723\)
Root analytic conductor: \(4.79723\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1033} (144, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1033,\ (0:\ ),\ -0.993 - 0.110i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03060163290 + 0.5513967726i\)
\(L(\frac12)\) \(\approx\) \(0.03060163290 + 0.5513967726i\)
\(L(1)\) \(\approx\) \(0.5760197840 + 0.3598841320i\)
\(L(1)\) \(\approx\) \(0.5760197840 + 0.3598841320i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1033 \( 1 \)
good2 \( 1 + (-0.894 + 0.446i)T \)
3 \( 1 + (0.676 + 0.736i)T \)
5 \( 1 + (-0.925 + 0.379i)T \)
7 \( 1 + (0.833 - 0.551i)T \)
11 \( 1 + (-0.970 + 0.241i)T \)
13 \( 1 + (0.0851 - 0.996i)T \)
17 \( 1 + (-0.541 + 0.840i)T \)
19 \( 1 + (0.859 + 0.510i)T \)
23 \( 1 + (-0.157 - 0.987i)T \)
29 \( 1 + (-0.561 + 0.827i)T \)
31 \( 1 + (-0.133 + 0.991i)T \)
37 \( 1 + (-0.694 + 0.719i)T \)
41 \( 1 + (-0.820 + 0.571i)T \)
43 \( 1 + (-0.847 - 0.531i)T \)
47 \( 1 + (0.760 + 0.648i)T \)
53 \( 1 + (0.791 + 0.611i)T \)
59 \( 1 + (-0.561 - 0.827i)T \)
61 \( 1 + (-0.997 - 0.0729i)T \)
67 \( 1 + (0.368 + 0.929i)T \)
71 \( 1 + (-0.925 - 0.379i)T \)
73 \( 1 + (0.694 + 0.719i)T \)
79 \( 1 + (0.276 - 0.961i)T \)
83 \( 1 + (-0.760 + 0.648i)T \)
89 \( 1 + (-0.639 + 0.768i)T \)
97 \( 1 + (-0.345 + 0.938i)T \)
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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

−20.92414922813606614084764240954, −20.215853276198592722652974807249, −19.60384544147218189160664267888, −18.636701152560593574758906219494, −18.45638974136235812087793673932, −17.55685849147200479332894618880, −16.55558205283398760465968207352, −15.54633546405262890930603923230, −15.250109821060443802751060114286, −13.80525092753770475744509390040, −13.22325925124465009728802720953, −12.0475129076684466374897953460, −11.69092209620004775101601598223, −11.03878558097137057075763917606, −9.55641863389345541543790433634, −8.94996732269643819883243731071, −8.248683126491743352165334635549, −7.53257874481514847251504338058, −7.04876125168653962736371305687, −5.56372308197072339138062385725, −4.31427343340525191321418412015, −3.29312067727551746249872769430, −2.35567184917247407821154153496, −1.56966380282652790198248221071, −0.28527240067951628037971270142, 1.43550264709821847253368396241, 2.654234420982119371187375043359, 3.56062348265185770096041265658, 4.721536208973268466872494958537, 5.4018808724182076705689744419, 6.87714392635878180469040648224, 7.72744038211102916910623109494, 8.16447275708700998837641707107, 8.79461851792662159677059126267, 10.2279132298850473197677848136, 10.49198856490114795492637821047, 11.12114359387779659997943178800, 12.30509977708921594311087512743, 13.60501046880154202821850215072, 14.51583100580430265632852263846, 15.08704942224873145433225321576, 15.642987829790435310687197799890, 16.365347865618956496183453779155, 17.23555345332904509807963585768, 18.2383592248003607832515331563, 18.69392278051521689433791694340, 19.8763498509304444648017360815, 20.23672734545474106703009396846, 20.738027330221966008522168151266, 21.92708664172278985901665781472

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