Properties

Label 2-3040-3040.2469-c0-0-7
Degree $2$
Conductor $3040$
Sign $-0.0980 + 0.995i$
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  + (0.995 − 0.0980i)2-s + (−0.591 − 1.42i)3-s + (0.980 − 0.195i)4-s + (0.923 + 0.382i)5-s + (−0.728 − 1.36i)6-s + (0.956 − 0.290i)8-s + (−0.983 + 0.983i)9-s + (0.956 + 0.290i)10-s + (0.636 − 1.53i)11-s + (−0.858 − 1.28i)12-s + (−0.871 + 0.360i)13-s − 1.54i·15-s + (0.923 − 0.382i)16-s + (−0.881 + 1.07i)18-s + (−0.923 + 0.382i)19-s + (0.980 + 0.195i)20-s + ⋯
L(s)  = 1  + (0.995 − 0.0980i)2-s + (−0.591 − 1.42i)3-s + (0.980 − 0.195i)4-s + (0.923 + 0.382i)5-s + (−0.728 − 1.36i)6-s + (0.956 − 0.290i)8-s + (−0.983 + 0.983i)9-s + (0.956 + 0.290i)10-s + (0.636 − 1.53i)11-s + (−0.858 − 1.28i)12-s + (−0.871 + 0.360i)13-s − 1.54i·15-s + (0.923 − 0.382i)16-s + (−0.881 + 1.07i)18-s + (−0.923 + 0.382i)19-s + (0.980 + 0.195i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0980 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0980 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3040\)    =    \(2^{5} \cdot 5 \cdot 19\)
Sign: $-0.0980 + 0.995i$
Analytic conductor: \(1.51715\)
Root analytic conductor: \(1.23172\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3040} (2469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3040,\ (\ :0),\ -0.0980 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.278110952\)
\(L(\frac12)\) \(\approx\) \(2.278110952\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.995 + 0.0980i)T \)
5 \( 1 + (-0.923 - 0.382i)T \)
19 \( 1 + (0.923 - 0.382i)T \)
good3 \( 1 + (0.591 + 1.42i)T + (-0.707 + 0.707i)T^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + (-0.636 + 1.53i)T + (-0.707 - 0.707i)T^{2} \)
13 \( 1 + (0.871 - 0.360i)T + (0.707 - 0.707i)T^{2} \)
17 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + (0.707 - 0.707i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-0.181 - 0.0750i)T + (0.707 + 0.707i)T^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 + (0.707 + 0.707i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-0.485 + 1.17i)T + (-0.707 - 0.707i)T^{2} \)
59 \( 1 + (-0.707 - 0.707i)T^{2} \)
61 \( 1 + (-0.425 - 1.02i)T + (-0.707 + 0.707i)T^{2} \)
67 \( 1 + (0.222 + 0.536i)T + (-0.707 + 0.707i)T^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (-0.707 + 0.707i)T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 + 1.76T + T^{2} \)
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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

−8.487708176140583781635012753380, −7.61030654236620077617222342396, −6.80406312888443355102108449420, −6.37368333728125234261981056868, −5.84439286291547322446231075282, −5.16567299242284353235951460220, −3.94690827185496782797755283028, −2.83219093818560667632241805101, −2.07663970071948216554124604677, −1.16390363369579485743407653631, 1.83127295650742479402453418029, 2.74571426635166214382461075074, 4.03565983223104378294260004056, 4.52208469473960809537495583879, 5.10668059358355639687028568841, 5.73360263075881568015546578641, 6.59958089837418608868769443494, 7.25652271990494696265771073360, 8.478603209946542941907883833874, 9.474417353973220637235325185747

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