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Properties

Label 1-4031-4031.181-r0-0-0
Degree $1$
Conductor $4031$
Sign $0.973 - 0.227i$
Analytic cond. $18.7198$
Root an. cond. $18.7198$
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  + (0.826 + 0.563i)2-s + (−0.988 + 0.149i)3-s + (0.365 + 0.930i)4-s + (0.0747 − 0.997i)5-s + (−0.900 − 0.433i)6-s + (−0.988 + 0.149i)7-s + (−0.222 + 0.974i)8-s + (0.955 − 0.294i)9-s + (0.623 − 0.781i)10-s + (−0.733 + 0.680i)11-s + (−0.5 − 0.866i)12-s + (−0.733 + 0.680i)13-s + (−0.900 − 0.433i)14-s + (0.0747 + 0.997i)15-s + (−0.733 + 0.680i)16-s + (−0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (0.826 + 0.563i)2-s + (−0.988 + 0.149i)3-s + (0.365 + 0.930i)4-s + (0.0747 − 0.997i)5-s + (−0.900 − 0.433i)6-s + (−0.988 + 0.149i)7-s + (−0.222 + 0.974i)8-s + (0.955 − 0.294i)9-s + (0.623 − 0.781i)10-s + (−0.733 + 0.680i)11-s + (−0.5 − 0.866i)12-s + (−0.733 + 0.680i)13-s + (−0.900 − 0.433i)14-s + (0.0747 + 0.997i)15-s + (−0.733 + 0.680i)16-s + (−0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4031\)    =    \(29 \cdot 139\)
Sign: $0.973 - 0.227i$
Analytic conductor: \(18.7198\)
Root analytic conductor: \(18.7198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4031} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4031,\ (0:\ ),\ 0.973 - 0.227i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5334756071 - 0.06135577822i\)
\(L(\frac12)\) \(\approx\) \(0.5334756071 - 0.06135577822i\)
\(L(1)\) \(\approx\) \(0.7503726022 + 0.3097025183i\)
\(L(1)\) \(\approx\) \(0.7503726022 + 0.3097025183i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 \)
139 \( 1 \)
good2 \( 1 + (0.826 + 0.563i)T \)
3 \( 1 + (-0.988 + 0.149i)T \)
5 \( 1 + (0.0747 - 0.997i)T \)
7 \( 1 + (-0.988 + 0.149i)T \)
11 \( 1 + (-0.733 + 0.680i)T \)
13 \( 1 + (-0.733 + 0.680i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.988 - 0.149i)T \)
23 \( 1 + (-0.900 - 0.433i)T \)
31 \( 1 + (0.0747 - 0.997i)T \)
37 \( 1 + (-0.733 - 0.680i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.0747 + 0.997i)T \)
47 \( 1 + (0.955 + 0.294i)T \)
53 \( 1 + (0.0747 - 0.997i)T \)
59 \( 1 + T \)
61 \( 1 + (0.365 - 0.930i)T \)
67 \( 1 + (0.955 - 0.294i)T \)
71 \( 1 + (-0.733 + 0.680i)T \)
73 \( 1 + (0.0747 + 0.997i)T \)
79 \( 1 + (-0.222 + 0.974i)T \)
83 \( 1 + (0.365 + 0.930i)T \)
89 \( 1 + (0.0747 - 0.997i)T \)
97 \( 1 + (0.365 + 0.930i)T \)
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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

−18.745650553295859343280332226, −17.91424643391257915872174466552, −17.22898927952234613810929846794, −16.21884765148526850004032547397, −15.68175498282413895961769564717, −15.240430097476932312152128168839, −14.19390695082448685208906942014, −13.49115911594637949880704145148, −13.1256522512470896343176210604, −12.08453826228600906248831941776, −11.91019964950830900838757733194, −10.75924787486793594727175233259, −10.431473248441090658121969192368, −10.09359599294701764508746551076, −9.0293131211021772143565415678, −7.62291703770008806052946799719, −6.95734494207193687871378425014, −6.44231037526397399953820267234, −5.675662715077775995513001190517, −5.21324851356503760323044928709, −4.15494182731588971764507474094, −3.41329969944531561179690257159, −2.65947988768230484537881838287, −2.00387830232284373061441176442, −0.58513492043545850845742200432, 0.197777784892633363605194148477, 1.872348353500631124163157728826, 2.465158948887335299573474437339, 4.00873208117884141145273446524, 4.19167569619178664985457231017, 5.05387377297025398129226574544, 5.656523613369288331257323651, 6.41733062415792716694238990562, 6.862204642649587175324716528615, 7.8361483705059224349917047483, 8.58906676314696425974638322903, 9.555019752444866481985789650970, 10.09492058468927282776076793003, 11.06980900623778095273774516933, 11.89019507753307643733404505928, 12.52724242113037173396376851118, 12.85708410776810055645347573693, 13.330631582837161396251456031271, 14.47089880651833117801688265883, 15.27077608701785294542139172718, 15.83713070977758894182202416691, 16.29807167725548334679789835189, 17.07711107685415686468005431244, 17.28830543903533495034704236147, 18.18441792771129200014626985950

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