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Properties

Label 1-2675-2675.129-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: $\chi_{2675} (129, \cdot )$
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

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

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