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Properties

Label 1-1033-1033.1005-r0-0-0
Degree $1$
Conductor $1033$
Sign $0.314 + 0.949i$
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  + (0.942 + 0.334i)2-s + (0.133 − 0.991i)3-s + (0.776 + 0.630i)4-s + (−0.121 + 0.992i)5-s + (0.457 − 0.889i)6-s + (0.181 − 0.983i)7-s + (0.520 + 0.853i)8-s + (−0.964 − 0.264i)9-s + (−0.446 + 0.894i)10-s + (0.311 + 0.950i)11-s + (0.728 − 0.685i)12-s + (−0.264 + 0.964i)13-s + (0.5 − 0.866i)14-s + (0.967 + 0.252i)15-s + (0.205 + 0.978i)16-s + (0.620 + 0.783i)17-s + ⋯
L(s)  = 1  + (0.942 + 0.334i)2-s + (0.133 − 0.991i)3-s + (0.776 + 0.630i)4-s + (−0.121 + 0.992i)5-s + (0.457 − 0.889i)6-s + (0.181 − 0.983i)7-s + (0.520 + 0.853i)8-s + (−0.964 − 0.264i)9-s + (−0.446 + 0.894i)10-s + (0.311 + 0.950i)11-s + (0.728 − 0.685i)12-s + (−0.264 + 0.964i)13-s + (0.5 − 0.866i)14-s + (0.967 + 0.252i)15-s + (0.205 + 0.978i)16-s + (0.620 + 0.783i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.314 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.314 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1033\)
Sign: $0.314 + 0.949i$
Analytic conductor: \(4.79723\)
Root analytic conductor: \(4.79723\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1033} (1005, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1033,\ (0:\ ),\ 0.314 + 0.949i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.066592379 + 1.492620242i\)
\(L(\frac12)\) \(\approx\) \(2.066592379 + 1.492620242i\)
\(L(1)\) \(\approx\) \(1.752581289 + 0.4324626200i\)
\(L(1)\) \(\approx\) \(1.752581289 + 0.4324626200i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1033 \( 1 \)
good2 \( 1 + (0.942 + 0.334i)T \)
3 \( 1 + (0.133 - 0.991i)T \)
5 \( 1 + (-0.121 + 0.992i)T \)
7 \( 1 + (0.181 - 0.983i)T \)
11 \( 1 + (0.311 + 0.950i)T \)
13 \( 1 + (-0.264 + 0.964i)T \)
17 \( 1 + (0.620 + 0.783i)T \)
19 \( 1 + (-0.847 + 0.531i)T \)
23 \( 1 + (0.0486 + 0.998i)T \)
29 \( 1 + (-0.954 - 0.299i)T \)
31 \( 1 + (-0.601 + 0.798i)T \)
37 \( 1 + (0.744 - 0.667i)T \)
41 \( 1 + (-0.711 - 0.702i)T \)
43 \( 1 + (0.915 + 0.402i)T \)
47 \( 1 + (-0.0243 - 0.999i)T \)
53 \( 1 + (0.920 - 0.391i)T \)
59 \( 1 + (0.954 - 0.299i)T \)
61 \( 1 + (0.976 - 0.217i)T \)
67 \( 1 + (0.571 - 0.820i)T \)
71 \( 1 + (0.121 + 0.992i)T \)
73 \( 1 + (-0.667 + 0.744i)T \)
79 \( 1 + (-0.978 - 0.205i)T \)
83 \( 1 + (0.999 + 0.0243i)T \)
89 \( 1 + (0.489 + 0.872i)T \)
97 \( 1 + (0.999 - 0.0121i)T \)
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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

−21.42505916839856512242675642539, −20.71850930781408486989518745840, −20.2632868503371514721040410095, −19.375840872635881281491392785820, −18.564015958102491862630003426438, −17.15323373079177019937575567252, −16.40597719840729737212515925331, −15.869811244302820580795172190054, −14.959515904213968417088878601030, −14.57281494837494838610396906493, −13.39125193783010254144432062989, −12.74577630368499820185952581217, −11.796820126672801108737666864524, −11.27249243829292745286430251124, −10.303825853433393934753610877448, −9.350743589180654751846458094213, −8.68888501179001457106523907286, −7.76260726275100421878668051807, −6.12523559326157499984312524149, −5.504145325205366934199099064830, −4.89416583600997418360567017501, −4.035005846568932689255397870149, −3.0575068234624283902114160760, −2.29673432490420690496448255937, −0.72689793517215064324756508492, 1.68551804067239244481516848287, 2.21711642666383031481309211906, 3.63755921815000526216234195477, 3.99940752397421257295652906762, 5.432423784773679349781878969278, 6.3747743334556999082971326998, 7.15276761935336322555882170598, 7.37808890017598639956020641260, 8.37822876515249649216292797541, 9.83277546307676006061218597515, 10.86731177494528745644579845639, 11.554747846229257481892243456627, 12.34883491038814702690069158557, 13.1272133401743427004836569565, 13.92362026317501011343273094994, 14.71642662318309667467520594873, 14.79402966789046324490791273293, 16.26063023161954154279283002182, 17.20484141992374983859899812720, 17.54061335457872472501073687217, 18.744463561175211708590021077723, 19.48491248021145588946323593269, 20.09501204808908212630298566306, 21.07106898840536896870157850562, 21.85139103421235682018744647708

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