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Properties

Label 1-2675-2675.394-r1-0-0
Degree $1$
Conductor $2675$
Sign $-0.937 + 0.347i$
Analytic cond. $287.468$
Root an. cond. $287.468$
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  + (−0.408 − 0.912i)2-s + (0.765 + 0.643i)3-s + (−0.666 + 0.745i)4-s + (0.275 − 0.961i)6-s + (0.482 + 0.875i)7-s + (0.952 + 0.303i)8-s + (0.171 + 0.985i)9-s + (−0.700 − 0.713i)11-s + (−0.989 + 0.141i)12-s + (0.924 + 0.381i)13-s + (0.602 − 0.798i)14-s + (−0.112 − 0.993i)16-s + (−0.835 + 0.548i)17-s + (0.829 − 0.558i)18-s + (−0.159 + 0.987i)19-s + ⋯
L(s)  = 1  + (−0.408 − 0.912i)2-s + (0.765 + 0.643i)3-s + (−0.666 + 0.745i)4-s + (0.275 − 0.961i)6-s + (0.482 + 0.875i)7-s + (0.952 + 0.303i)8-s + (0.171 + 0.985i)9-s + (−0.700 − 0.713i)11-s + (−0.989 + 0.141i)12-s + (0.924 + 0.381i)13-s + (0.602 − 0.798i)14-s + (−0.112 − 0.993i)16-s + (−0.835 + 0.548i)17-s + (0.829 − 0.558i)18-s + (−0.159 + 0.987i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2675 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.937 + 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2675 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.937 + 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2675\)    =    \(5^{2} \cdot 107\)
Sign: $-0.937 + 0.347i$
Analytic conductor: \(287.468\)
Root analytic conductor: \(287.468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2675} (394, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2675,\ (1:\ ),\ -0.937 + 0.347i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1992796165 + 1.110069626i\)
\(L(\frac12)\) \(\approx\) \(0.1992796165 + 1.110069626i\)
\(L(1)\) \(\approx\) \(0.9880596703 + 0.1213133024i\)
\(L(1)\) \(\approx\) \(0.9880596703 + 0.1213133024i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
107 \( 1 \)
good2 \( 1 + (-0.408 - 0.912i)T \)
3 \( 1 + (0.765 + 0.643i)T \)
7 \( 1 + (0.482 + 0.875i)T \)
11 \( 1 + (-0.700 - 0.713i)T \)
13 \( 1 + (0.924 + 0.381i)T \)
17 \( 1 + (-0.835 + 0.548i)T \)
19 \( 1 + (-0.159 + 0.987i)T \)
23 \( 1 + (-0.945 + 0.325i)T \)
29 \( 1 + (0.620 - 0.783i)T \)
31 \( 1 + (-0.543 + 0.839i)T \)
37 \( 1 + (0.611 - 0.791i)T \)
41 \( 1 + (0.878 - 0.477i)T \)
43 \( 1 + (-0.320 - 0.947i)T \)
47 \( 1 + (0.991 - 0.130i)T \)
53 \( 1 + (0.994 + 0.106i)T \)
59 \( 1 + (0.963 + 0.269i)T \)
61 \( 1 + (0.124 + 0.992i)T \)
67 \( 1 + (-0.592 + 0.805i)T \)
71 \( 1 + (-0.997 + 0.0710i)T \)
73 \( 1 + (0.772 - 0.634i)T \)
79 \( 1 + (-0.974 - 0.223i)T \)
83 \( 1 + (-0.639 + 0.768i)T \)
89 \( 1 + (-0.342 + 0.939i)T \)
97 \( 1 + (-0.749 + 0.661i)T \)
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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

−18.47415172061084535744888021535, −18.02978128986472780902752645809, −17.71004324564031826702329693420, −16.73681795351188295114674880386, −15.81921785417141460687126527906, −15.3613885730669097728857035661, −14.532922170967466242101325209937, −13.965048341320851012881007153783, −13.14887809836266580695966844253, −12.98219681014741128025039286785, −11.53313051935328487297862593657, −10.73570549617950300731108503544, −9.96120227417928663681794751499, −9.195969646871129999641930639728, −8.375364266453728362875381203957, −7.903522111033489254229235028133, −7.153105226202082245523926961392, −6.65890165648637987239250187764, −5.7312905839397661848311463863, −4.60180622128355002600757441695, −4.145456165745180344083884578676, −2.86296452506506293855448072409, −1.89491097556738718573006829836, −0.98461336417358466526510966527, −0.201084227130816172839962350733, 1.28655700169141205192456881852, 2.209461185475306047539687562525, 2.63900920868274271815725157946, 3.87695814416900634228442874060, 4.05189364575673624386742224343, 5.297527688274192856964832205842, 5.94591504416299442905848586606, 7.454905096783824581339948645005, 8.273793063867559787207339150243, 8.63791687313614994535901871015, 9.18486289489024466219722736048, 10.24606984622835156104772824878, 10.66739381607527071987435987733, 11.4172554921590596165244635534, 12.1685943196391089480676281177, 13.07546101221607211961527410649, 13.719687923262318702438227682221, 14.314602660171642265578012256188, 15.24005434274643648136641569517, 16.031067980497634475080109506602, 16.43280737292226316794784905653, 17.63360763003489528113534007906, 18.20200263592307967801616213052, 18.91097328675314314699476228423, 19.389690888635934211171958462136

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