↑

Properties

Label 1-31-31.4-r0-0-0
Degree $1$
Conductor $31$
Sign $0.938 - 0.344i$
Analytic cond. $0.143963$
Root an. cond. $0.143963$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.809 + 0.587i)2-s + (−0.809 − 0.587i)3-s + (0.309 − 0.951i)4-s + 5-s + 6-s + (0.309 − 0.951i)7-s + (0.309 + 0.951i)8-s + (0.309 + 0.951i)9-s + (−0.809 + 0.587i)10-s + (0.309 − 0.951i)11-s + (−0.809 + 0.587i)12-s + (−0.809 − 0.587i)13-s + (0.309 + 0.951i)14-s + (−0.809 − 0.587i)15-s + (−0.809 − 0.587i)16-s + (0.309 + 0.951i)17-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)2-s + (−0.809 − 0.587i)3-s + (0.309 − 0.951i)4-s + 5-s + 6-s + (0.309 − 0.951i)7-s + (0.309 + 0.951i)8-s + (0.309 + 0.951i)9-s + (−0.809 + 0.587i)10-s + (0.309 − 0.951i)11-s + (−0.809 + 0.587i)12-s + (−0.809 − 0.587i)13-s + (0.309 + 0.951i)14-s + (−0.809 − 0.587i)15-s + (−0.809 − 0.587i)16-s + (0.309 + 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(31\)
Sign: $0.938 - 0.344i$
Analytic conductor: \(0.143963\)
Root analytic conductor: \(0.143963\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{31} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 31,\ (0:\ ),\ 0.938 - 0.344i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4872922874 - 0.08663360516i\)
\(L(\frac12)\) \(\approx\) \(0.4872922874 - 0.08663360516i\)
\(L(1)\) \(\approx\) \(0.6421325000 - 0.04173340352i\)
\(L(1)\) \(\approx\) \(0.6421325000 - 0.04173340352i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 \)
good2 \( 1 + (-0.809 + 0.587i)T \)
3 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + T \)
7 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (0.309 - 0.951i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (0.309 + 0.951i)T \)
29 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + (-0.809 + 0.587i)T \)
47 \( 1 + (-0.809 - 0.587i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.809 - 0.587i)T \)
61 \( 1 + T \)
67 \( 1 + T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (-0.809 + 0.587i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (0.309 - 0.951i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−37.02420809917010949759578280776, −35.78461935097228262905727223825, −34.28731005008986926683828489531, −33.73226990366241923104722202862, −32.01622051544968760125337756521, −30.33059559418254464220399650667, −29.041523476379104022022642646701, −28.33617249557216725704055846710, −27.26551561345988570455062122635, −25.87374903615421744602828687152, −24.672691645708714148085354762182, −22.43771995697331436509591399382, −21.57286622791687329283535667168, −20.58795544117813723002443248797, −18.61858504682483825202115459698, −17.5926699844629935506970913278, −16.66545824457283164567124076965, −14.95999181704614635613125877264, −12.60826987827210059159471693928, −11.49069172082830781192186822586, −9.95826353494920462909999372513, −9.121132365908701768376826295388, −6.71385282733999819961242558602, −4.8323168868215700743181173804, −2.230737701503585425079311641070, 1.465747569970922076813830119683, 5.3918364152072317095315675634, 6.57188145630580090324180777719, 8.01973491542183046046375517198, 10.01980136177853379636217216437, 11.06259935081869934911009093260, 13.18808881244569304076336796826, 14.552497716056245332993842749643, 16.80181970668045869197347914128, 17.140188846639522519924703360658, 18.42935342094157269445976140789, 19.75512661348436705919202148334, 21.64539995405185955593021348848, 23.30020804679606300728919413267, 24.3318726341242740444724393903, 25.363917920329099615485899278393, 26.85298352198849416729105728598, 28.000281992509842645352197202442, 29.51244333135626553230778423672, 29.7675633177331052689002550423, 32.3959811081114960178155525809, 33.45920675798276877585999131737, 34.32427543544962154636425630936, 35.50367286191600893468726671662, 36.66818438259717001471596286753

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