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Properties

Label 1-1033-1033.113-r0-0-0
Degree $1$
Conductor $1033$
Sign $0.917 + 0.398i$
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  + (−0.457 − 0.889i)2-s + (−0.694 + 0.719i)3-s + (−0.581 + 0.813i)4-s + (0.744 − 0.667i)5-s + (0.957 + 0.288i)6-s + (−0.457 + 0.889i)7-s + (0.989 + 0.145i)8-s + (−0.0365 − 0.999i)9-s + (−0.934 − 0.357i)10-s + (−0.322 + 0.946i)11-s + (−0.181 − 0.983i)12-s + (−0.0365 − 0.999i)13-s + 14-s + (−0.0365 + 0.999i)15-s + (−0.322 − 0.946i)16-s + (0.639 + 0.768i)17-s + ⋯
L(s)  = 1  + (−0.457 − 0.889i)2-s + (−0.694 + 0.719i)3-s + (−0.581 + 0.813i)4-s + (0.744 − 0.667i)5-s + (0.957 + 0.288i)6-s + (−0.457 + 0.889i)7-s + (0.989 + 0.145i)8-s + (−0.0365 − 0.999i)9-s + (−0.934 − 0.357i)10-s + (−0.322 + 0.946i)11-s + (−0.181 − 0.983i)12-s + (−0.0365 − 0.999i)13-s + 14-s + (−0.0365 + 0.999i)15-s + (−0.322 − 0.946i)16-s + (0.639 + 0.768i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.917 + 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.917 + 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1033\)
Sign: $0.917 + 0.398i$
Analytic conductor: \(4.79723\)
Root analytic conductor: \(4.79723\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1033} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1033,\ (0:\ ),\ 0.917 + 0.398i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8037455189 + 0.1670746365i\)
\(L(\frac12)\) \(\approx\) \(0.8037455189 + 0.1670746365i\)
\(L(1)\) \(\approx\) \(0.6994069806 - 0.05653841258i\)
\(L(1)\) \(\approx\) \(0.6994069806 - 0.05653841258i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1033 \( 1 \)
good2 \( 1 + (-0.457 - 0.889i)T \)
3 \( 1 + (-0.694 + 0.719i)T \)
5 \( 1 + (0.744 - 0.667i)T \)
7 \( 1 + (-0.457 + 0.889i)T \)
11 \( 1 + (-0.322 + 0.946i)T \)
13 \( 1 + (-0.0365 - 0.999i)T \)
17 \( 1 + (0.639 + 0.768i)T \)
19 \( 1 + (-0.976 - 0.217i)T \)
23 \( 1 + (0.957 + 0.288i)T \)
29 \( 1 + (0.252 + 0.967i)T \)
31 \( 1 + (0.744 - 0.667i)T \)
37 \( 1 + (-0.322 - 0.946i)T \)
41 \( 1 + (-0.0365 + 0.999i)T \)
43 \( 1 + (-0.791 - 0.611i)T \)
47 \( 1 + (0.989 + 0.145i)T \)
53 \( 1 + (0.744 - 0.667i)T \)
59 \( 1 + (0.252 - 0.967i)T \)
61 \( 1 + (0.252 + 0.967i)T \)
67 \( 1 + (-0.872 + 0.489i)T \)
71 \( 1 + (0.744 + 0.667i)T \)
73 \( 1 + (-0.322 + 0.946i)T \)
79 \( 1 + (-0.322 + 0.946i)T \)
83 \( 1 + (0.989 - 0.145i)T \)
89 \( 1 + (-0.997 + 0.0729i)T \)
97 \( 1 + (-0.997 - 0.0729i)T \)
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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

−21.73129655392700538663518333764, −20.84984001569255402914466984285, −19.26534819075959361636894864052, −19.07915893550522164686093544838, −18.378443785310698834379278542821, −17.48006534454474819022648473221, −16.76384911673569017263131190633, −16.52947509394915151174363872979, −15.35618249092989418089222060530, −14.1829048127854549064992823380, −13.721913344436350944567016775572, −13.23637497315981149184345709647, −11.913801134243172892683730718784, −10.7735882334680340344247058203, −10.43557646784844062720199512205, −9.45060998542790861283845076895, −8.40768098139611198721613905556, −7.4148849369807679554046395135, −6.724596411506691074924578826188, −6.286247008507203550737939442055, −5.39631920073691990944188301598, −4.40657513531709562195631504652, −2.92004814643108011361995779225, −1.63981269522787236327791045062, −0.58785082394945818376798198574, 0.930901797805205552004624941219, 2.13592412803512259574711164584, 3.06308521247372701498538969648, 4.18810076851049701821789927196, 5.13468375050113936812627898541, 5.67464127759606524563348081335, 6.88228294127987680521595104608, 8.330718081770604465580886636149, 8.93806686734803152890792006986, 9.8774684662159799111232534894, 10.17710404692448551712685192449, 11.102855964884063098715968487525, 12.23802409632976186134077843390, 12.62696838850712365190823690026, 13.19726531386627690859343753376, 14.724398209141841253167055240396, 15.42942168069320499015059063903, 16.40642163599714931750038961656, 17.155288853529297840272810398223, 17.66298471912588578602496379857, 18.35780569546963141390700807329, 19.36508981751706077571055208679, 20.25350687309240019629306857738, 20.975229350368999038665952057240, 21.48218891287665610293465795040

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