Properties

Label 2-2520-280.37-c0-0-2
Degree $2$
Conductor $2520$
Sign $0.417 - 0.908i$
Analytic cond. $1.25764$
Root an. cond. $1.12144$
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  + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (0.258 − 0.965i)5-s + (0.5 + 0.866i)7-s + (−0.707 − 0.707i)8-s + 10-s + (0.448 − 0.258i)11-s + (−0.707 + 0.707i)14-s + (0.500 − 0.866i)16-s + (0.258 + 0.965i)20-s + (0.366 + 0.366i)22-s + (−0.866 − 0.499i)25-s + (−0.866 − 0.5i)28-s + 1.93·29-s + (0.866 + 1.5i)31-s + (0.965 + 0.258i)32-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (0.258 − 0.965i)5-s + (0.5 + 0.866i)7-s + (−0.707 − 0.707i)8-s + 10-s + (0.448 − 0.258i)11-s + (−0.707 + 0.707i)14-s + (0.500 − 0.866i)16-s + (0.258 + 0.965i)20-s + (0.366 + 0.366i)22-s + (−0.866 − 0.499i)25-s + (−0.866 − 0.5i)28-s + 1.93·29-s + (0.866 + 1.5i)31-s + (0.965 + 0.258i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.417 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.417 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.417 - 0.908i$
Analytic conductor: \(1.25764\)
Root analytic conductor: \(1.12144\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :0),\ 0.417 - 0.908i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.418647850\)
\(L(\frac12)\) \(\approx\) \(1.418647850\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
3 \( 1 \)
5 \( 1 + (-0.258 + 0.965i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good11 \( 1 + (-0.448 + 0.258i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (-0.866 - 0.5i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T^{2} \)
29 \( 1 - 1.93T + T^{2} \)
31 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.866 + 0.5i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (0.866 - 0.5i)T^{2} \)
53 \( 1 + (-0.448 + 1.67i)T + (-0.866 - 0.5i)T^{2} \)
59 \( 1 + (0.258 + 0.448i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.866 + 0.5i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
79 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.366 + 0.366i)T + iT^{2} \)
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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

−8.927805939118164253701186198860, −8.425266045743047748414688975382, −7.948069171947057717129367179930, −6.69731143440582654644501230860, −6.20098260333993653579199049125, −5.14485435626499306822652987479, −4.92041274712850289776835967107, −3.86916830248358277741343485253, −2.68321378210836323913352064999, −1.21354967044264094691340569144, 1.14382914784148481329963440074, 2.29417936912339871320025238626, 3.12822044867230113948935167455, 4.13028352869219741151580845356, 4.65667046599941111823052389951, 5.85082888384187495588323170786, 6.54740378402180785845623114776, 7.46368774638921253951332817347, 8.245383348345092408853837144472, 9.200876031606048797439481930439

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