↑

Properties

Label 1-507-507.125-r0-0-0
Degree $1$
Conductor $507$
Sign $0.872 + 0.489i$
Analytic cond. $2.35449$
Root an. cond. $2.35449$
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  + (−0.464 − 0.885i)2-s + (−0.568 + 0.822i)4-s + (0.935 + 0.354i)5-s + (−0.992 − 0.120i)7-s + (0.992 + 0.120i)8-s + (−0.120 − 0.992i)10-s + (−0.464 + 0.885i)11-s + (0.354 + 0.935i)14-s + (−0.354 − 0.935i)16-s + (0.120 − 0.992i)17-s + i·19-s + (−0.822 + 0.568i)20-s + 22-s + 23-s + (0.748 + 0.663i)25-s + ⋯
L(s)  = 1  + (−0.464 − 0.885i)2-s + (−0.568 + 0.822i)4-s + (0.935 + 0.354i)5-s + (−0.992 − 0.120i)7-s + (0.992 + 0.120i)8-s + (−0.120 − 0.992i)10-s + (−0.464 + 0.885i)11-s + (0.354 + 0.935i)14-s + (−0.354 − 0.935i)16-s + (0.120 − 0.992i)17-s + i·19-s + (−0.822 + 0.568i)20-s + 22-s + 23-s + (0.748 + 0.663i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.872 + 0.489i$
Analytic conductor: \(2.35449\)
Root analytic conductor: \(2.35449\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 507,\ (0:\ ),\ 0.872 + 0.489i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8550201881 + 0.2234328469i\)
\(L(\frac12)\) \(\approx\) \(0.8550201881 + 0.2234328469i\)
\(L(1)\) \(\approx\) \(0.8056375762 - 0.09754539808i\)
\(L(1)\) \(\approx\) \(0.8056375762 - 0.09754539808i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.464 - 0.885i)T \)
5 \( 1 + (0.935 + 0.354i)T \)
7 \( 1 + (-0.992 - 0.120i)T \)
11 \( 1 + (-0.464 + 0.885i)T \)
17 \( 1 + (0.120 - 0.992i)T \)
19 \( 1 + iT \)
23 \( 1 + T \)
29 \( 1 + (-0.885 + 0.464i)T \)
31 \( 1 + (-0.663 - 0.748i)T \)
37 \( 1 + (0.663 + 0.748i)T \)
41 \( 1 + (0.239 + 0.970i)T \)
43 \( 1 + (0.748 + 0.663i)T \)
47 \( 1 + (-0.822 + 0.568i)T \)
53 \( 1 + (-0.120 + 0.992i)T \)
59 \( 1 + (-0.935 - 0.354i)T \)
61 \( 1 + (0.120 + 0.992i)T \)
67 \( 1 + (-0.822 + 0.568i)T \)
71 \( 1 + (0.239 + 0.970i)T \)
73 \( 1 + (0.464 - 0.885i)T \)
79 \( 1 + (0.568 + 0.822i)T \)
83 \( 1 + (-0.239 + 0.970i)T \)
89 \( 1 + iT \)
97 \( 1 + (0.935 - 0.354i)T \)
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

−23.81988438018718736253483196535, −22.8063050278786403473405960099, −21.930048826603688343157816714433, −21.20964494811269113294290514255, −19.89236693524733970313058689449, −19.148601323125880740830824945456, −18.3814141964682188916737012932, −17.40751839449723482525786964078, −16.75299209235210095395581015439, −16.02658347652667676270727534767, −15.15962964932170930701970842327, −14.12129600930891057015383600815, −13.21893387661251674858434422588, −12.79976381100003531436724450395, −10.97506078784620240999256599428, −10.24781509407462185435355064755, −9.17128463143610005930433281618, −8.82470999190502486215525443647, −7.51608800020499663222094184164, −6.45924072443183907050447635508, −5.80803691126497269122381592326, −4.984346628448900805179225413626, −3.48936778829180766275184905375, −2.07431704817812824834550395698, −0.59266976406849305467451989900, 1.3239987753679460040557611961, 2.51677473401042558506995906132, 3.20545628588989243515220193003, 4.5314791868655721541226376712, 5.707894683422299266705252255422, 6.93349834515968877000368471984, 7.75348226913209361213294423917, 9.31105226180952095927245685369, 9.60125454508515591408437011109, 10.44663342287672886831277677432, 11.34690822336827921486253801246, 12.63070316483675784070975750148, 13.02881182007332474047861650219, 13.97060251609787259568142405358, 15.02579580783906744156553360964, 16.41505894752526366378411524788, 16.93240489683851400375152136014, 18.108494958574347034882731445474, 18.480193676968407371597650346449, 19.42341211675408341398414043414, 20.49897911861051549619318499273, 20.888911122864975491880297824546, 22.01015150839482821871025133780, 22.63342208074618242685560132399, 23.23956660628324530109823401495

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