↑

Properties

Label 1-2675-2675.1031-r1-0-0
Degree $1$
Conductor $2675$
Sign $0.172 - 0.985i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.353 − 0.935i)2-s + (−0.135 + 0.990i)3-s + (−0.749 + 0.661i)4-s + (0.974 − 0.223i)6-s + (−0.889 − 0.456i)7-s + (0.884 + 0.467i)8-s + (−0.963 − 0.269i)9-s + (0.503 − 0.864i)11-s + (−0.553 − 0.832i)12-s + (−0.342 − 0.939i)13-s + (−0.112 + 0.993i)14-s + (0.124 − 0.992i)16-s + (−0.419 − 0.907i)17-s + (0.0887 + 0.996i)18-s + (0.997 + 0.0710i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.935i)2-s + (−0.135 + 0.990i)3-s + (−0.749 + 0.661i)4-s + (0.974 − 0.223i)6-s + (−0.889 − 0.456i)7-s + (0.884 + 0.467i)8-s + (−0.963 − 0.269i)9-s + (0.503 − 0.864i)11-s + (−0.553 − 0.832i)12-s + (−0.342 − 0.939i)13-s + (−0.112 + 0.993i)14-s + (0.124 − 0.992i)16-s + (−0.419 − 0.907i)17-s + (0.0887 + 0.996i)18-s + (0.997 + 0.0710i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2675 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.172 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2675 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.172 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2675\)    =    \(5^{2} \cdot 107\)
Sign: $0.172 - 0.985i$
Analytic conductor: \(287.468\)
Root analytic conductor: \(287.468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2675} (1031, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2675,\ (1:\ ),\ 0.172 - 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8057910309 - 0.6770248943i\)
\(L(\frac12)\) \(\approx\) \(0.8057910309 - 0.6770248943i\)
\(L(1)\) \(\approx\) \(0.6734150625 - 0.1899998837i\)
\(L(1)\) \(\approx\) \(0.6734150625 - 0.1899998837i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
107 \( 1 \)
good2 \( 1 + (-0.353 - 0.935i)T \)
3 \( 1 + (-0.135 + 0.990i)T \)
7 \( 1 + (-0.889 - 0.456i)T \)
11 \( 1 + (0.503 - 0.864i)T \)
13 \( 1 + (-0.342 - 0.939i)T \)
17 \( 1 + (-0.419 - 0.907i)T \)
19 \( 1 + (0.997 + 0.0710i)T \)
23 \( 1 + (-0.979 - 0.200i)T \)
29 \( 1 + (0.543 + 0.839i)T \)
31 \( 1 + (0.822 + 0.568i)T \)
37 \( 1 + (-0.229 - 0.973i)T \)
41 \( 1 + (-0.297 + 0.954i)T \)
43 \( 1 + (-0.0296 + 0.999i)T \)
47 \( 1 + (0.801 + 0.597i)T \)
53 \( 1 + (0.998 + 0.0474i)T \)
59 \( 1 + (-0.0533 - 0.998i)T \)
61 \( 1 + (-0.451 + 0.892i)T \)
67 \( 1 + (-0.440 + 0.897i)T \)
71 \( 1 + (0.472 - 0.881i)T \)
73 \( 1 + (-0.461 - 0.886i)T \)
79 \( 1 + (-0.969 + 0.246i)T \)
83 \( 1 + (0.952 + 0.303i)T \)
89 \( 1 + (0.657 + 0.753i)T \)
97 \( 1 + (-0.523 + 0.851i)T \)
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

−19.074271061412836170184739140060, −18.58362779631800836971884439958, −17.80866103607727646153999864039, −17.12583300133696961691043705707, −16.73413242729228990103424885328, −15.64072379391976199281933356108, −15.26806268603665577678186153649, −14.209540867226413325645922109697, −13.72290313407460169171585629285, −13.045299950931554886694608527344, −12.02706456508982471378636109509, −11.83715255444153591331256871229, −10.35328371684945291266354995914, −9.75101671885897433508240376237, −8.96463565586233984675921039154, −8.34198993246809041585282144421, −7.37204627863521088369880218713, −6.88899682120172097146250107432, −6.21472936354899562933628819570, −5.65967698491197804483617116547, −4.58986532407051546284775069828, −3.72412431341539003178621083307, −2.35407624394726192609338303987, −1.68013440196830677839009905519, −0.564258835863788773993920078701, 0.38650634055158832151057845881, 1.045255607292485268970183359819, 2.70659790898845108234951290974, 3.10710880370270835817124961663, 3.82314687248621101822770090259, 4.65215022494835311824997954760, 5.459471727025160781120974020108, 6.379831360530037176520617628671, 7.47630340178116447307443810232, 8.3475905115434096516692337102, 9.15761103840447033094924188729, 9.65867517370311487471323928261, 10.37326218068519410577286586975, 10.846354395728446468919558612756, 11.7765720185136885294709564880, 12.23559003987662312020897135183, 13.30160234230791794691195582916, 13.90453950704079629363765875198, 14.528846853057118773576333995962, 15.82753206264389437668027985672, 16.18104980856755244143260024421, 16.79814125340965005168487657535, 17.728799204797847410324026144709, 18.13242023859979296730167070667, 19.2918210403591211075621006411

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