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Properties

Label 1-6041-6041.377-r0-0-0
Degree $1$
Conductor $6041$
Sign $-0.555 - 0.831i$
Analytic cond. $28.0542$
Root an. cond. $28.0542$
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  + (0.393 − 0.919i)2-s + (0.624 + 0.781i)3-s + (−0.690 − 0.723i)4-s + (−0.290 + 0.956i)5-s + (0.963 − 0.266i)6-s + (−0.936 + 0.349i)8-s + (−0.220 + 0.975i)9-s + (0.765 + 0.644i)10-s + (0.926 − 0.376i)11-s + (0.134 − 0.990i)12-s + (−0.814 + 0.580i)13-s + (−0.929 + 0.370i)15-s + (−0.0473 + 0.998i)16-s + (−0.999 + 0.0437i)17-s + (0.809 + 0.586i)18-s + (−0.641 + 0.767i)19-s + ⋯
L(s)  = 1  + (0.393 − 0.919i)2-s + (0.624 + 0.781i)3-s + (−0.690 − 0.723i)4-s + (−0.290 + 0.956i)5-s + (0.963 − 0.266i)6-s + (−0.936 + 0.349i)8-s + (−0.220 + 0.975i)9-s + (0.765 + 0.644i)10-s + (0.926 − 0.376i)11-s + (0.134 − 0.990i)12-s + (−0.814 + 0.580i)13-s + (−0.929 + 0.370i)15-s + (−0.0473 + 0.998i)16-s + (−0.999 + 0.0437i)17-s + (0.809 + 0.586i)18-s + (−0.641 + 0.767i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6041\)    =    \(7 \cdot 863\)
Sign: $-0.555 - 0.831i$
Analytic conductor: \(28.0542\)
Root analytic conductor: \(28.0542\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6041} (377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6041,\ (0:\ ),\ -0.555 - 0.831i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2530620053 - 0.4737073216i\)
\(L(\frac12)\) \(\approx\) \(0.2530620053 - 0.4737073216i\)
\(L(1)\) \(\approx\) \(1.060182171 - 0.06304143055i\)
\(L(1)\) \(\approx\) \(1.060182171 - 0.06304143055i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
863 \( 1 \)
good2 \( 1 + (0.393 - 0.919i)T \)
3 \( 1 + (0.624 + 0.781i)T \)
5 \( 1 + (-0.290 + 0.956i)T \)
11 \( 1 + (0.926 - 0.376i)T \)
13 \( 1 + (-0.814 + 0.580i)T \)
17 \( 1 + (-0.999 + 0.0437i)T \)
19 \( 1 + (-0.641 + 0.767i)T \)
23 \( 1 + (-0.0837 + 0.996i)T \)
29 \( 1 + (-0.700 - 0.713i)T \)
31 \( 1 + (0.234 - 0.972i)T \)
37 \( 1 + (-0.998 - 0.0510i)T \)
41 \( 1 + (-0.255 + 0.966i)T \)
43 \( 1 + (-0.206 - 0.978i)T \)
47 \( 1 + (0.783 - 0.621i)T \)
53 \( 1 + (-0.413 - 0.910i)T \)
59 \( 1 + (0.386 - 0.922i)T \)
61 \( 1 + (-0.842 - 0.538i)T \)
67 \( 1 + (-0.965 - 0.259i)T \)
71 \( 1 + (0.497 + 0.867i)T \)
73 \( 1 + (0.684 + 0.728i)T \)
79 \( 1 + (-0.993 - 0.116i)T \)
83 \( 1 + (0.982 - 0.188i)T \)
89 \( 1 + (0.406 - 0.913i)T \)
97 \( 1 + (0.459 - 0.888i)T \)
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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

−17.67991840865105035981933533881, −17.28731196699620598717285958422, −16.70158991060184551127747340981, −15.837031115943183661450224154716, −15.16093486438253850869063283440, −14.78288706557778452501248356754, −13.95619247167424092484490562499, −13.43411870379903395920923827155, −12.619333291094712766729758923936, −12.41759108529089405140975161154, −11.79923075926844102909729495517, −10.65471932108104666143476888868, −9.47975325363759766080441222877, −8.824323178742329657053326855398, −8.73546039585040364814806020308, −7.7443158308976353941609957229, −7.1830432506544469296723535865, −6.62002000659976636672264676320, −5.89025304618206495498769578205, −4.87319077611594639200849544974, −4.46708464474910406254966945731, −3.64904831904839448220833879848, −2.78903148234574896073326516455, −1.91592603581194632371540740523, −0.8839011530673576384464161220, 0.11635920540360701998599356877, 1.913160078484122768298651406807, 2.08650612008027614549242326446, 3.10990917734853544514213734758, 3.77180848875992819552110765608, 4.10730627357893720643486657370, 4.92503484134245356880156369066, 5.86626149407901233719402602036, 6.55920480522846426321269426215, 7.468161433616555404808089232241, 8.35173233079816414726342562895, 9.03126271240174493247089606529, 9.714780556943586239840576576137, 10.13949797116481815633542487496, 10.95678004668587117818547165711, 11.48351854281337478226216011112, 11.91175954879829213638182915931, 12.98668910014444496721085646764, 13.75371685671628640061152758131, 14.13514535495584152518415645324, 14.839425467449312618435927071916, 15.189803738486344224832329547578, 15.8871162127730265451022740922, 17.0517915723286222876175252531, 17.32597289257294238813474110292

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