↑

Properties

Label 2-1519-1519.743-c0-0-1
Degree $2$
Conductor $1519$
Sign $-0.838 + 0.545i$
Analytic cond. $0.758079$
Root an. cond. $0.870677$
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  + (−1.12 − 1.40i)2-s + (−0.500 + 2.19i)4-s + (0.400 − 0.193i)5-s + (−0.222 − 0.974i)7-s + (2.02 − 0.974i)8-s + (0.623 − 0.781i)9-s + (−0.722 − 0.347i)10-s + (−1.12 + 1.40i)14-s + (−1.62 − 0.781i)16-s − 1.80·18-s + 1.24·19-s + (0.222 + 0.974i)20-s + (−0.499 + 0.626i)25-s + 2.24·28-s + 31-s + (0.222 + 0.974i)32-s + ⋯
L(s)  = 1  + (−1.12 − 1.40i)2-s + (−0.500 + 2.19i)4-s + (0.400 − 0.193i)5-s + (−0.222 − 0.974i)7-s + (2.02 − 0.974i)8-s + (0.623 − 0.781i)9-s + (−0.722 − 0.347i)10-s + (−1.12 + 1.40i)14-s + (−1.62 − 0.781i)16-s − 1.80·18-s + 1.24·19-s + (0.222 + 0.974i)20-s + (−0.499 + 0.626i)25-s + 2.24·28-s + 31-s + (0.222 + 0.974i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.838 + 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.838 + 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1519\)    =    \(7^{2} \cdot 31\)
Sign: $-0.838 + 0.545i$
Analytic conductor: \(0.758079\)
Root analytic conductor: \(0.870677\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1519} (743, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1519,\ (\ :0),\ -0.838 + 0.545i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6503048537\)
\(L(\frac12)\) \(\approx\) \(0.6503048537\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.222 + 0.974i)T \)
31 \( 1 - T \)
good2 \( 1 + (1.12 + 1.40i)T + (-0.222 + 0.974i)T^{2} \)
3 \( 1 + (-0.623 + 0.781i)T^{2} \)
5 \( 1 + (-0.400 + 0.193i)T + (0.623 - 0.781i)T^{2} \)
11 \( 1 + (0.222 - 0.974i)T^{2} \)
13 \( 1 + (0.222 - 0.974i)T^{2} \)
17 \( 1 + (0.900 - 0.433i)T^{2} \)
19 \( 1 - 1.24T + T^{2} \)
23 \( 1 + (0.900 + 0.433i)T^{2} \)
29 \( 1 + (0.900 - 0.433i)T^{2} \)
37 \( 1 + (0.900 - 0.433i)T^{2} \)
41 \( 1 + (-0.400 + 0.193i)T + (0.623 - 0.781i)T^{2} \)
43 \( 1 + (-0.623 - 0.781i)T^{2} \)
47 \( 1 + (1.12 + 1.40i)T + (-0.222 + 0.974i)T^{2} \)
53 \( 1 + (0.900 + 0.433i)T^{2} \)
59 \( 1 + (1.12 + 0.541i)T + (0.623 + 0.781i)T^{2} \)
61 \( 1 + (0.900 - 0.433i)T^{2} \)
67 \( 1 + 0.445T + T^{2} \)
71 \( 1 + (-0.400 + 1.75i)T + (-0.900 - 0.433i)T^{2} \)
73 \( 1 + (0.222 + 0.974i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (0.222 + 0.974i)T^{2} \)
89 \( 1 + (0.222 + 0.974i)T^{2} \)
97 \( 1 + 0.445T + T^{2} \)
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

−9.596146415354940441426411251378, −8.946453901751373748320991532896, −7.88303782429219928065348650261, −7.29407506937087969391599450004, −6.29746958006978414556508906648, −4.83376006536107591377801451906, −3.76021671707593864530907943650, −3.17436200866204424233264220599, −1.77211116161650999392942663038, −0.858525597436016161348465712067, 1.48414308378752047556159135684, 2.76526983332398996052432469948, 4.59195170661938374896495012409, 5.44018577982595340500170048607, 6.09685811318007784410276323291, 6.83555993590744844332982889583, 7.72988719727334899831358243629, 8.211758747189303500290762272753, 9.159315903253028636763022468789, 9.740618531134729373272649053784

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