Properties

Label 1-837-837.407-r1-0-0
Degree $1$
Conductor $837$
Sign $0.995 - 0.0957i$
Analytic cond. $89.9481$
Root an. cond. $89.9481$
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  + (−0.961 + 0.275i)2-s + (0.848 − 0.529i)4-s + (−0.173 + 0.984i)5-s + (−0.374 − 0.927i)7-s + (−0.669 + 0.743i)8-s + (−0.104 − 0.994i)10-s + (−0.990 − 0.139i)11-s + (0.961 + 0.275i)13-s + (0.615 + 0.788i)14-s + (0.438 − 0.898i)16-s + (0.978 + 0.207i)17-s + (0.913 + 0.406i)19-s + (0.374 + 0.927i)20-s + (0.990 − 0.139i)22-s + (0.374 − 0.927i)23-s + ⋯
L(s)  = 1  + (−0.961 + 0.275i)2-s + (0.848 − 0.529i)4-s + (−0.173 + 0.984i)5-s + (−0.374 − 0.927i)7-s + (−0.669 + 0.743i)8-s + (−0.104 − 0.994i)10-s + (−0.990 − 0.139i)11-s + (0.961 + 0.275i)13-s + (0.615 + 0.788i)14-s + (0.438 − 0.898i)16-s + (0.978 + 0.207i)17-s + (0.913 + 0.406i)19-s + (0.374 + 0.927i)20-s + (0.990 − 0.139i)22-s + (0.374 − 0.927i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.995 - 0.0957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.995 - 0.0957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.995 - 0.0957i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ 0.995 - 0.0957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.010517294 - 0.04849805995i\)
\(L(\frac12)\) \(\approx\) \(1.010517294 - 0.04849805995i\)
\(L(1)\) \(\approx\) \(0.6655314772 + 0.08852695697i\)
\(L(1)\) \(\approx\) \(0.6655314772 + 0.08852695697i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.961 + 0.275i)T \)
5 \( 1 + (-0.173 + 0.984i)T \)
7 \( 1 + (-0.374 - 0.927i)T \)
11 \( 1 + (-0.990 - 0.139i)T \)
13 \( 1 + (0.961 + 0.275i)T \)
17 \( 1 + (0.978 + 0.207i)T \)
19 \( 1 + (0.913 + 0.406i)T \)
23 \( 1 + (0.374 - 0.927i)T \)
29 \( 1 + (-0.961 + 0.275i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.559 + 0.829i)T \)
43 \( 1 + (-0.719 - 0.694i)T \)
47 \( 1 + (0.997 - 0.0697i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (0.719 - 0.694i)T \)
61 \( 1 + (0.766 - 0.642i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (0.978 + 0.207i)T \)
73 \( 1 + (-0.978 + 0.207i)T \)
79 \( 1 + (0.0348 - 0.999i)T \)
83 \( 1 + (-0.961 + 0.275i)T \)
89 \( 1 + (0.978 - 0.207i)T \)
97 \( 1 + (0.990 + 0.139i)T \)
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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

−21.57305349706960594475709514200, −20.94092114423561851441221642554, −20.41477200262577752258336380616, −19.47839645356018988488318439101, −18.69605405103904741393904149644, −18.11772267798856739353079664113, −17.22943652852276418586207710529, −16.25165190497890216868599934535, −15.79987774182232570511683523224, −15.179149240315756559013734797639, −13.51458356947945816284092808579, −12.836003406730094770466595553186, −12.045479307661553986097680817255, −11.37334764459255459752993703219, −10.29221255112532768169285422390, −9.39632959856328528395695463432, −8.85377340307308410409514813098, −7.96088508248165824195099484835, −7.28127012629591347719459010419, −5.78126176785526519947769948723, −5.33975142528831397234098299241, −3.70443310721260031956474671140, −2.85533544683259947668007729595, −1.68507528394956376711018095673, −0.68226401791494523423836901006, 0.47693911478421876683707494220, 1.64191948584534425724320432317, 2.98046573205310371449825027195, 3.63215838880710415229474093284, 5.27888526822738931586594014190, 6.26446445083512054508090209046, 7.02677054715861713331926744223, 7.73489852350653833442253128874, 8.4943896920919158178009925594, 9.824147996143835028641963444720, 10.310389160919803143135250156233, 10.98604821977684066369093040478, 11.791521729215976169046914242429, 13.1112005527845987852531046913, 14.0370154846927700105860540075, 14.765694122927675829161381578881, 15.73591031917785335956780565455, 16.342257887078304006582926003078, 17.09164628792522456475758715945, 18.171348730705610741609197188723, 18.683895940839446207766846811765, 19.16886648626971612450957578587, 20.50359748075589379634070642081, 20.62637514068058654843501789991, 21.94449245290843926435417833328

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