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Properties

Label 1-837-837.272-r0-0-0
Degree $1$
Conductor $837$
Sign $0.995 + 0.0956i$
Analytic cond. $3.88701$
Root an. cond. $3.88701$
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 + (0.939 − 0.342i)5-s + (0.990 + 0.139i)7-s + (−0.669 + 0.743i)8-s + (−0.809 + 0.587i)10-s + (−0.374 − 0.927i)11-s + (0.241 + 0.970i)13-s + (−0.990 + 0.139i)14-s + (0.438 − 0.898i)16-s + (0.669 − 0.743i)17-s + (−0.809 + 0.587i)19-s + (0.615 − 0.788i)20-s + (0.615 + 0.788i)22-s + (−0.374 + 0.927i)23-s + ⋯
L(s)  = 1  + (−0.961 + 0.275i)2-s + (0.848 − 0.529i)4-s + (0.939 − 0.342i)5-s + (0.990 + 0.139i)7-s + (−0.669 + 0.743i)8-s + (−0.809 + 0.587i)10-s + (−0.374 − 0.927i)11-s + (0.241 + 0.970i)13-s + (−0.990 + 0.139i)14-s + (0.438 − 0.898i)16-s + (0.669 − 0.743i)17-s + (−0.809 + 0.587i)19-s + (0.615 − 0.788i)20-s + (0.615 + 0.788i)22-s + (−0.374 + 0.927i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.995 + 0.0956i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (272, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ 0.995 + 0.0956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.308977071 + 0.06272982273i\)
\(L(\frac12)\) \(\approx\) \(1.308977071 + 0.06272982273i\)
\(L(1)\) \(\approx\) \(0.9597817256 + 0.04667932987i\)
\(L(1)\) \(\approx\) \(0.9597817256 + 0.04667932987i\)

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 \( 1 + (0.939 - 0.342i)T \)
7 \( 1 + (0.990 + 0.139i)T \)
11 \( 1 + (-0.374 - 0.927i)T \)
13 \( 1 + (0.241 + 0.970i)T \)
17 \( 1 + (0.669 - 0.743i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (-0.374 + 0.927i)T \)
29 \( 1 + (0.961 - 0.275i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (0.997 + 0.0697i)T \)
43 \( 1 + (-0.961 + 0.275i)T \)
47 \( 1 + (0.997 - 0.0697i)T \)
53 \( 1 + (0.669 - 0.743i)T \)
59 \( 1 + (0.241 + 0.970i)T \)
61 \( 1 + (-0.766 + 0.642i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (-0.669 - 0.743i)T \)
79 \( 1 + (-0.848 - 0.529i)T \)
83 \( 1 + (-0.719 - 0.694i)T \)
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

−21.67933171823147737500917793431, −21.301719791551694911314751499457, −20.43752842528848990413907761143, −19.87875184101492680506153929319, −18.623545342933129585990994483915, −18.12310451654155405796228726662, −17.40147779397279443916292128337, −17.0184754215994238275520489746, −15.715878486788223584647268794874, −14.938372452345259413297269845625, −14.21107932539415463602430040369, −12.90264895895999041405401262048, −12.41100540869465074252813631671, −11.12118811055281862114104144147, −10.468535499796464138834704522050, −10.02908761498626654707016302797, −8.85772804713676935547897161178, −8.12189550704482818964258620921, −7.29197527490627781003217358230, −6.326133763296656304961745335691, −5.37481321408790289971634708362, −4.16118009040429345527805088816, −2.73065535259713192616794885544, −2.078034906042168375495715857818, −1.04942473605270413535647725092, 1.068204817903956836946934003567, 1.85725609466855173252023105067, 2.85088614158494892235445924809, 4.51887298448954183978387089116, 5.59537096326462089058024352408, 6.11416143612085923359945925109, 7.27736403482996896674523007202, 8.2743935140179306592568047365, 8.78831516636145951290958557537, 9.73741386682866635428715537908, 10.48123586915522774203480163335, 11.42239318598306443384814532492, 12.051265090674565491149562025641, 13.516891625180643105737583646096, 14.12589101495956303039566198982, 14.910135228615913844986453375918, 16.07399277497871958950907045673, 16.633817743746120757902475374756, 17.37078611749928338024726784543, 18.18964320405166473105283365497, 18.68896911777200964880981444829, 19.60711500604620614713929847826, 20.73239108861813419885293987425, 21.18835431515370714320865251511, 21.699273582267264771084222110350

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