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Properties

Label 2-3040-3040.2469-c0-0-2
Degree $2$
Conductor $3040$
Sign $0.634 + 0.773i$
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  + (−0.634 + 0.773i)2-s + (−0.761 − 1.83i)3-s + (−0.195 − 0.980i)4-s + (−0.923 − 0.382i)5-s + (1.90 + 0.577i)6-s + (0.881 + 0.471i)8-s + (−2.09 + 2.09i)9-s + (0.881 − 0.471i)10-s + (−0.425 + 1.02i)11-s + (−1.65 + 1.10i)12-s + (1.76 − 0.732i)13-s + 1.99i·15-s + (−0.923 + 0.382i)16-s + (−0.290 − 2.94i)18-s + (0.923 − 0.382i)19-s + (−0.195 + 0.980i)20-s + ⋯
L(s)  = 1  + (−0.634 + 0.773i)2-s + (−0.761 − 1.83i)3-s + (−0.195 − 0.980i)4-s + (−0.923 − 0.382i)5-s + (1.90 + 0.577i)6-s + (0.881 + 0.471i)8-s + (−2.09 + 2.09i)9-s + (0.881 − 0.471i)10-s + (−0.425 + 1.02i)11-s + (−1.65 + 1.10i)12-s + (1.76 − 0.732i)13-s + 1.99i·15-s + (−0.923 + 0.382i)16-s + (−0.290 − 2.94i)18-s + (0.923 − 0.382i)19-s + (−0.195 + 0.980i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.634 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.634 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3040\)    =    \(2^{5} \cdot 5 \cdot 19\)
Sign: $0.634 + 0.773i$
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: \((2,\ 3040,\ (\ :0),\ 0.634 + 0.773i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5447589213\)
\(L(\frac12)\) \(\approx\) \(0.5447589213\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.634 - 0.773i)T \)
5 \( 1 + (0.923 + 0.382i)T \)
19 \( 1 + (-0.923 + 0.382i)T \)
good3 \( 1 + (0.761 + 1.83i)T + (-0.707 + 0.707i)T^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + (0.425 - 1.02i)T + (-0.707 - 0.707i)T^{2} \)
13 \( 1 + (-1.76 + 0.732i)T + (0.707 - 0.707i)T^{2} \)
17 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + (0.707 - 0.707i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-1.42 - 0.591i)T + (0.707 + 0.707i)T^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 + (0.707 + 0.707i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (0.0750 - 0.181i)T + (-0.707 - 0.707i)T^{2} \)
59 \( 1 + (-0.707 - 0.707i)T^{2} \)
61 \( 1 + (-0.636 - 1.53i)T + (-0.707 + 0.707i)T^{2} \)
67 \( 1 + (0.360 + 0.871i)T + (-0.707 + 0.707i)T^{2} \)
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 \( 1 + 0.580T + T^{2} \)
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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

−8.348089022644211656480812814207, −7.87584265132694273093091371556, −7.40218567605663062476126298689, −6.72424995686711870912459551053, −5.95381281395542512151601277358, −5.36217698685093026391277251531, −4.47464756921752836221533359978, −2.85692441778528103186226559464, −1.50237810312977994272914693728, −0.807037847904901476122377726525, 0.78328784429321338777483151989, 2.89357736058999935334140994805, 3.69479320398835847888327842171, 3.88275861732806277778839331272, 4.89611203997275863954376491272, 5.86731992245014642298779141204, 6.62057110949919508909405200270, 7.919393956986898076986998417837, 8.569841839124860644529063425440, 9.095263969293070132263730391044

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