↑

Properties

Label 2-2527-133.118-c0-0-6
Degree $2$
Conductor $2527$
Sign $-0.919 + 0.392i$
Analytic cond. $1.26113$
Root an. cond. $1.12300$
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  + (−0.939 − 0.342i)2-s + (−0.984 + 0.173i)3-s + (−0.642 − 0.766i)5-s + (0.984 + 0.173i)6-s + (−0.866 − 0.5i)7-s + (0.499 + 0.866i)8-s + (0.342 + 0.939i)10-s + (0.984 + 0.173i)13-s + (0.642 + 0.766i)14-s + (0.766 + 0.642i)15-s + (−0.173 − 0.984i)16-s + (0.342 − 0.939i)17-s + (0.939 + 0.342i)21-s + (0.766 + 0.642i)23-s + (−0.642 − 0.766i)24-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)2-s + (−0.984 + 0.173i)3-s + (−0.642 − 0.766i)5-s + (0.984 + 0.173i)6-s + (−0.866 − 0.5i)7-s + (0.499 + 0.866i)8-s + (0.342 + 0.939i)10-s + (0.984 + 0.173i)13-s + (0.642 + 0.766i)14-s + (0.766 + 0.642i)15-s + (−0.173 − 0.984i)16-s + (0.342 − 0.939i)17-s + (0.939 + 0.342i)21-s + (0.766 + 0.642i)23-s + (−0.642 − 0.766i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2527 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.919 + 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2527 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.919 + 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2527\)    =    \(7 \cdot 19^{2}\)
Sign: $-0.919 + 0.392i$
Analytic conductor: \(1.26113\)
Root analytic conductor: \(1.12300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2527} (1182, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2527,\ (\ :0),\ -0.919 + 0.392i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2461885821\)
\(L(\frac12)\) \(\approx\) \(0.2461885821\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 \)
good2 \( 1 + (0.939 + 0.342i)T + (0.766 + 0.642i)T^{2} \)
3 \( 1 + (0.984 - 0.173i)T + (0.939 - 0.342i)T^{2} \)
5 \( 1 + (0.642 + 0.766i)T + (-0.173 + 0.984i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.984 - 0.173i)T + (0.939 + 0.342i)T^{2} \)
17 \( 1 + (-0.342 + 0.939i)T + (-0.766 - 0.642i)T^{2} \)
23 \( 1 + (-0.766 - 0.642i)T + (0.173 + 0.984i)T^{2} \)
29 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.984 - 0.173i)T + (0.939 - 0.342i)T^{2} \)
43 \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \)
47 \( 1 + (0.342 + 0.939i)T + (-0.766 + 0.642i)T^{2} \)
53 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
59 \( 1 + (0.342 - 0.939i)T + (-0.766 - 0.642i)T^{2} \)
61 \( 1 + (-0.642 + 0.766i)T + (-0.173 - 0.984i)T^{2} \)
67 \( 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2} \)
71 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
73 \( 1 + (-0.984 + 0.173i)T + (0.939 - 0.342i)T^{2} \)
79 \( 1 + (-0.173 - 0.984i)T + (-0.939 + 0.342i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.984 + 0.173i)T + (0.939 + 0.342i)T^{2} \)
97 \( 1 + (-0.342 + 0.939i)T + (-0.766 - 0.642i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.768853938083125538499287890014, −8.300220180986514179000638742544, −7.32937149171160764407697699354, −6.49563752117117398947310079281, −5.53852685666321271162211423319, −4.89639711208032403099502824130, −4.05261192278712778209408609929, −2.90875765863441594502505553286, −1.21692970588049221477211577481, −0.35511506587193515755598761223, 1.11995550485380537265076748626, 3.01227824631214274974846979917, 3.64301498052237234732135869572, 4.79915673631575832325754509850, 6.01762311852949244926840916023, 6.41189692051548303896105145799, 7.06465294485212957963934511245, 7.956773968823906627610703311125, 8.644532544892137144958541769479, 9.251490808291178872412153489943

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