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Properties

Label 2-3040-760.499-c0-0-0
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: $\chi_{3040} (879, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3040,\ (\ :0),\ -0.0389 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3907837024\)
\(L(\frac12)\) \(\approx\) \(0.3907837024\)
\(L(1)\) 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

−9.291309250356182227356132350584, −8.227898381732497263525084906421, −7.48800207086286526992936998963, −7.05941430549065263942940554259, −6.18685934312982656953661143590, −4.96260173276290046878965673524, −4.49407184417684513471998114624, −3.62128983192008867641820725788, −2.74095144944991728544369950658, −1.23882706076437555797219738064, 0.25425026458404651203876292289, 2.46613039674785034014087840901, 3.16101283851270357471416363230, 3.45566123964525480743267198578, 5.22602337365417515655670357151, 5.64591124352035806529197715428, 6.21842398413124960905718883878, 7.33579850468883186326643604595, 8.155676758090623993596030218437, 8.505563815186817545965185871965

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