↑

Properties

Label 2-1520-380.379-c1-0-12
Degree $2$
Conductor $1520$
Sign $-0.287 - 0.957i$
Analytic cond. $12.1372$
Root an. cond. $3.48385$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.55i·3-s + (−2.03 − 0.928i)5-s + 1.46·7-s + 0.593·9-s − 2.19i·11-s − 3.35·13-s + (1.44 − 3.15i)15-s + 3.77i·17-s + (−4.27 + 0.851i)19-s + 2.26i·21-s + 7.11·23-s + (3.27 + 3.77i)25-s + 5.57i·27-s + 6.65i·29-s + 1.93·31-s + ⋯
L(s)  = 1  + 0.895i·3-s + (−0.909 − 0.415i)5-s + 0.552·7-s + 0.197·9-s − 0.661i·11-s − 0.931·13-s + (0.372 − 0.814i)15-s + 0.916i·17-s + (−0.980 + 0.195i)19-s + 0.495i·21-s + 1.48·23-s + (0.654 + 0.755i)25-s + 1.07i·27-s + 1.23i·29-s + 0.346·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.287 - 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.287 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $-0.287 - 0.957i$
Analytic conductor: \(12.1372\)
Root analytic conductor: \(3.48385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (1519, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :1/2),\ -0.287 - 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.208944529\)
\(L(\frac12)\) \(\approx\) \(1.208944529\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (2.03 + 0.928i)T \)
19 \( 1 + (4.27 - 0.851i)T \)
good3 \( 1 - 1.55iT - 3T^{2} \)
7 \( 1 - 1.46T + 7T^{2} \)
11 \( 1 + 2.19iT - 11T^{2} \)
13 \( 1 + 3.35T + 13T^{2} \)
17 \( 1 - 3.77iT - 17T^{2} \)
23 \( 1 - 7.11T + 23T^{2} \)
29 \( 1 - 6.65iT - 29T^{2} \)
31 \( 1 - 1.93T + 31T^{2} \)
37 \( 1 - 4.72T + 37T^{2} \)
41 \( 1 - 7.88iT - 41T^{2} \)
43 \( 1 - 3.49T + 43T^{2} \)
47 \( 1 + 5.37T + 47T^{2} \)
53 \( 1 - 10.9T + 53T^{2} \)
59 \( 1 + 13.4T + 59T^{2} \)
61 \( 1 + 3.80T + 61T^{2} \)
67 \( 1 - 0.493iT - 67T^{2} \)
71 \( 1 + 1.93T + 71T^{2} \)
73 \( 1 - 11.0iT - 73T^{2} \)
79 \( 1 + 0.949T + 79T^{2} \)
83 \( 1 - 3.79T + 83T^{2} \)
89 \( 1 - 18.3iT - 89T^{2} \)
97 \( 1 - 0.649T + 97T^{2} \)
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

−9.638293607082353012905647010514, −8.878473829009369579367543889688, −8.234696791940134968118404596417, −7.44996145450290754760882039771, −6.49849966550345111415725187567, −5.20992588204956731989139311234, −4.65636108053331267183840430105, −3.92742825643501138419798693933, −2.95560316596776006259717114615, −1.28925175777824011607726910756, 0.52277526845661624651651874960, 2.01581029113973870043303630680, 2.89670322849723984942048142466, 4.35958931088749207908499930937, 4.80502632656531359988899164949, 6.23099811219978275796832732756, 7.16610007370644960117158902260, 7.39068863582212460533981423945, 8.166604718376395585573523538859, 9.143373027578073201545538131500

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