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Properties

Label 1-1032-1032.251-r0-0-0
Degree $1$
Conductor $1032$
Sign $-0.953 + 0.300i$
Analytic cond. $4.79258$
Root an. cond. $4.79258$
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  + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)7-s − 11-s + (0.5 + 0.866i)13-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + (−0.5 − 0.866i)29-s + (0.5 − 0.866i)31-s − 35-s + (0.5 − 0.866i)37-s − 41-s + 47-s + (−0.5 + 0.866i)49-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)7-s − 11-s + (0.5 + 0.866i)13-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + (−0.5 − 0.866i)29-s + (0.5 − 0.866i)31-s − 35-s + (0.5 − 0.866i)37-s − 41-s + 47-s + (−0.5 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1032 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.953 + 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1032 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.953 + 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1032\)    =    \(2^{3} \cdot 3 \cdot 43\)
Sign: $-0.953 + 0.300i$
Analytic conductor: \(4.79258\)
Root analytic conductor: \(4.79258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1032} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1032,\ (0:\ ),\ -0.953 + 0.300i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1327223815 + 0.8637999601i\)
\(L(\frac12)\) \(\approx\) \(0.1327223815 + 0.8637999601i\)
\(L(1)\) \(\approx\) \(0.7914469050 + 0.3866469910i\)
\(L(1)\) \(\approx\) \(0.7914469050 + 0.3866469910i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
43 \( 1 \)
good5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 - T \)
13 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 - T \)
47 \( 1 + T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 - T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + T \)
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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

−20.91723292632039076325212828517, −20.4181791031784186285732591985, −20.00179725209459785908423279527, −18.84192193123560621701433494787, −18.07142653140762199597876576898, −17.284460155875482805731692085096, −16.460959776408678350127195952253, −15.82655057373605249872011144930, −15.06056304592660593182934640095, −13.98070128495351153783851345227, −13.24222125224263942268965777640, −12.60737346052153283969296819600, −11.643201874184877741992401942205, −10.7713020496222502230145404039, −10.15631662583790980713016585662, −8.95115058652957888301587099534, −8.1410319107691886659998364420, −7.628665386526456643095538471181, −6.58636146526216959625207029911, −5.18501632068343903404657423015, −4.85530002157325899802645000606, −3.75520961515109929151188033914, −2.76419799391017288597822004731, −1.34554701288655571076959420304, −0.37571197608953041070318776956, 1.72645977171919566997846073094, 2.499191076884488405829134483870, 3.615450119850500989334360655376, 4.406075769261540403342699702244, 5.77157853440682344202319834852, 6.156064406401571084969189361525, 7.572356148663542230571741617, 7.97357493315794990581028590339, 8.94854827871426906711613592807, 10.05110423252736319838479212570, 10.78284015991467517444110005330, 11.614868288712848378150567844387, 12.18927478302332573060566640206, 13.28237991867323694843961540706, 14.15999932211598360832187010146, 14.99999493436370695370686556890, 15.45295406925342773215824151804, 16.32419651177737754853075523146, 17.34405928839425262540623015156, 18.263981163883243846021946265124, 18.83174158288922845376943109884, 19.237618374192976424072997756322, 20.5292859026630631379446351382, 21.27833726379981701516454817919, 21.77547201259757542357086727424

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