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Properties

Label 1-1061-1061.29-r1-0-0
Degree $1$
Conductor $1061$
Sign $0.967 + 0.252i$
Analytic cond. $114.020$
Root an. cond. $114.020$
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  + (−0.587 − 0.809i)2-s + (−0.951 − 0.309i)3-s + (−0.309 + 0.951i)4-s + (−0.809 − 0.587i)5-s + (0.309 + 0.951i)6-s + (−0.809 + 0.587i)7-s + (0.951 − 0.309i)8-s + (0.809 + 0.587i)9-s + i·10-s + (−0.309 + 0.951i)11-s + (0.587 − 0.809i)12-s + i·13-s + (0.951 + 0.309i)14-s + (0.587 + 0.809i)15-s + (−0.809 − 0.587i)16-s − i·17-s + ⋯
L(s)  = 1  + (−0.587 − 0.809i)2-s + (−0.951 − 0.309i)3-s + (−0.309 + 0.951i)4-s + (−0.809 − 0.587i)5-s + (0.309 + 0.951i)6-s + (−0.809 + 0.587i)7-s + (0.951 − 0.309i)8-s + (0.809 + 0.587i)9-s + i·10-s + (−0.309 + 0.951i)11-s + (0.587 − 0.809i)12-s + i·13-s + (0.951 + 0.309i)14-s + (0.587 + 0.809i)15-s + (−0.809 − 0.587i)16-s − i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1061 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1061 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1061\)
Sign: $0.967 + 0.252i$
Analytic conductor: \(114.020\)
Root analytic conductor: \(114.020\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1061} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1061,\ (1:\ ),\ 0.967 + 0.252i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3711650602 + 0.04754367122i\)
\(L(\frac12)\) \(\approx\) \(0.3711650602 + 0.04754367122i\)
\(L(1)\) \(\approx\) \(0.3994632673 - 0.1353694803i\)
\(L(1)\) \(\approx\) \(0.3994632673 - 0.1353694803i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1061 \( 1 \)
good2 \( 1 + (-0.587 - 0.809i)T \)
3 \( 1 + (-0.951 - 0.309i)T \)
5 \( 1 + (-0.809 - 0.587i)T \)
7 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (-0.309 + 0.951i)T \)
13 \( 1 + iT \)
17 \( 1 - iT \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (-0.309 - 0.951i)T \)
29 \( 1 + (0.587 - 0.809i)T \)
31 \( 1 - T \)
37 \( 1 + T \)
41 \( 1 + (-0.309 - 0.951i)T \)
43 \( 1 + (0.587 - 0.809i)T \)
47 \( 1 + (0.309 + 0.951i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + iT \)
61 \( 1 + iT \)
67 \( 1 + (-0.309 - 0.951i)T \)
71 \( 1 + (0.587 + 0.809i)T \)
73 \( 1 + iT \)
79 \( 1 + (-0.587 - 0.809i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.951 - 0.309i)T \)
97 \( 1 + (0.809 - 0.587i)T \)
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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

−21.71664729836442597320510997051, −20.035696266460054091747297886, −19.682529805704388080332525928886, −18.72003144533357122354772734134, −18.07630036046461872845600959585, −17.30178176780979197116456853506, −16.50089367907609529478356467639, −15.8683094558784402773076466651, −15.42003345585032527979534342890, −14.515114138142286605182033934542, −13.37566452090731162832008139230, −12.66064322973349549223036950948, −11.28860867401931111622049905283, −10.84239364828179675006573970697, −10.18102105686094135452473146743, −9.3084425953096735624040569227, −8.119937460084355776590204825897, −7.4318489894710816528711175368, −6.5686203862746824067097043655, −5.95917193296480871493801268327, −5.037776178276920625828772046009, −3.93392103713000093367517101323, −3.10607283646905600184430153802, −1.07591073475388138663382013282, −0.25568356732185042453508063916, 0.515978302151764303375464414095, 1.69947115942990444425587456843, 2.65558037504655114622033158730, 4.03510037020800512352890983788, 4.59772191282846897395292500305, 5.719678147062589764809618928570, 6.96831262595688933363125126898, 7.510779109403383798931323545480, 8.57586833323029315897762090608, 9.4921142797832644751384682031, 10.10001932885228647305514696298, 11.19386778127461807610202506303, 11.905629096665871092705822081545, 12.384573383885446942602111791032, 12.873625773150515896125030252341, 13.987256917920423148215433327567, 15.490604293240323942171904949934, 16.26796676720351789226236380914, 16.5623511649533718645394882902, 17.570670329100422352591011263699, 18.557514136550808425557793211, 18.767419771697007242126430859095, 19.68975133561218876984911875825, 20.51687652330426779162816526158, 21.18591413282756639360040454225

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