Properties

Label 2-2520-7.2-c1-0-2
Degree $2$
Conductor $2520$
Sign $-0.788 - 0.615i$
Analytic cond. $20.1223$
Root an. cond. $4.48578$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)5-s + (0.835 + 2.51i)7-s + (−3.11 − 5.39i)11-s − 3.77·13-s + (−0.313 − 0.542i)17-s + (−0.206 + 0.357i)19-s + (−2.04 + 3.53i)23-s + (−0.499 − 0.866i)25-s + 4.12·29-s + (4.32 + 7.49i)31-s + (2.59 + 0.532i)35-s + (−3.59 + 6.22i)37-s − 4.88·41-s − 10.6·43-s + (−5.49 + 9.52i)47-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (0.315 + 0.948i)7-s + (−0.938 − 1.62i)11-s − 1.04·13-s + (−0.0759 − 0.131i)17-s + (−0.0474 + 0.0821i)19-s + (−0.425 + 0.737i)23-s + (−0.0999 − 0.173i)25-s + 0.766·29-s + (0.777 + 1.34i)31-s + (0.438 + 0.0899i)35-s + (−0.590 + 1.02i)37-s − 0.763·41-s − 1.62·43-s + (−0.802 + 1.38i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 - 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.788 - 0.615i$
Analytic conductor: \(20.1223\)
Root analytic conductor: \(4.48578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (1801, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :1/2),\ -0.788 - 0.615i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4517135512\)
\(L(\frac12)\) \(\approx\) \(0.4517135512\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.835 - 2.51i)T \)
good11 \( 1 + (3.11 + 5.39i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.77T + 13T^{2} \)
17 \( 1 + (0.313 + 0.542i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.206 - 0.357i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.04 - 3.53i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.12T + 29T^{2} \)
31 \( 1 + (-4.32 - 7.49i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.59 - 6.22i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.88T + 41T^{2} \)
43 \( 1 + 10.6T + 43T^{2} \)
47 \( 1 + (5.49 - 9.52i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.271 - 0.470i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.16 - 7.21i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.963 + 1.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.50 + 6.07i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.3T + 71T^{2} \)
73 \( 1 + (2.58 + 4.47i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.57 - 6.19i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.0T + 83T^{2} \)
89 \( 1 + (-1.60 + 2.78i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 13.7T + 97T^{2} \)
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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

−9.097344882604655199770324476643, −8.262989632981353746224322883185, −8.111625877318152304655269268182, −6.81701752919050881923100729265, −6.00606935149957435636334285470, −5.19974672613106551845836158248, −4.82725405705475356865534531200, −3.27332544926560389335648829279, −2.67769014197661838164826881530, −1.44730267266190528269439831205, 0.14006408836172163893677899466, 1.89049520780728281273025617133, 2.56792915062809416001204961660, 3.86869475425694790695853941139, 4.71887405507961198147933898769, 5.22857810608549133020265356556, 6.59089149011807981171007902948, 7.05678547385874437420006294356, 7.76871159633980044785389148755, 8.411231077660840635775047924187

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