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Properties

Label 2-3040-760.99-c0-0-1
Degree $2$
Conductor $3040$
Sign $-0.0389 + 0.999i$
Analytic cond. $1.51715$
Root an. cond. $1.23172$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.766 − 0.642i)5-s + (0.766 − 1.32i)7-s + (−0.939 + 0.342i)9-s + (−0.939 − 1.62i)11-s + (−0.173 + 0.984i)13-s + (0.939 + 0.342i)19-s + (−0.266 − 0.223i)23-s + (0.173 − 0.984i)25-s + (−0.266 − 1.50i)35-s + 0.347·37-s + (−0.326 − 1.85i)41-s + (−0.5 + 0.866i)45-s + (−0.939 + 0.342i)47-s + (−0.673 − 1.16i)49-s + (1.17 + 0.984i)53-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)5-s + (0.766 − 1.32i)7-s + (−0.939 + 0.342i)9-s + (−0.939 − 1.62i)11-s + (−0.173 + 0.984i)13-s + (0.939 + 0.342i)19-s + (−0.266 − 0.223i)23-s + (0.173 − 0.984i)25-s + (−0.266 − 1.50i)35-s + 0.347·37-s + (−0.326 − 1.85i)41-s + (−0.5 + 0.866i)45-s + (−0.939 + 0.342i)47-s + (−0.673 − 1.16i)49-s + (1.17 + 0.984i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0389 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0389 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3040\)    =    \(2^{5} \cdot 5 \cdot 19\)
Sign: $-0.0389 + 0.999i$
Analytic conductor: \(1.51715\)
Root analytic conductor: \(1.23172\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3040} (1999, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3040,\ (\ :0),\ -0.0389 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.283576315\)
\(L(\frac12)\) \(\approx\) \(1.283576315\)
\(L(1)\) 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 \( 1 + (-0.766 + 0.642i)T \)
19 \( 1 + (-0.939 - 0.342i)T \)
good3 \( 1 + (0.939 - 0.342i)T^{2} \)
7 \( 1 + (-0.766 + 1.32i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.939 + 1.62i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
17 \( 1 + (-0.766 - 0.642i)T^{2} \)
23 \( 1 + (0.266 + 0.223i)T + (0.173 + 0.984i)T^{2} \)
29 \( 1 + (-0.766 + 0.642i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - 0.347T + T^{2} \)
41 \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \)
43 \( 1 + (-0.173 + 0.984i)T^{2} \)
47 \( 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2} \)
53 \( 1 + (-1.17 - 0.984i)T + (0.173 + 0.984i)T^{2} \)
59 \( 1 + (0.939 + 0.342i)T + (0.766 + 0.642i)T^{2} \)
61 \( 1 + (-0.173 - 0.984i)T^{2} \)
67 \( 1 + (-0.766 + 0.642i)T^{2} \)
71 \( 1 + (-0.173 + 0.984i)T^{2} \)
73 \( 1 + (0.939 - 0.342i)T^{2} \)
79 \( 1 + (0.939 - 0.342i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2} \)
97 \( 1 + (-0.766 - 0.642i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.634274994326864181555531677541, −8.040372957764707799912439155570, −7.38301311434810973425863949403, −6.28525436236880085429473393517, −5.55051530904007454597745996651, −4.99292076794807458579569145435, −4.05763911789926881279356050319, −3.03468110327645499546955445172, −1.93927786235634076111140753869, −0.77383324126801386941563652194, 1.79108073477511873239768111000, 2.59857289482275635214955875936, 3.11933602333262775517039295237, 4.79494576040320461209077361086, 5.33881415191275298543205370536, 5.85457488962895464714455699010, 6.81334860056388656915593088441, 7.72005411528345613503314620570, 8.246816976926065541630586078055, 9.197674659148863189845754087911

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