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Properties

Label 2-2520-7.4-c1-0-3
Degree $2$
Conductor $2520$
Sign $-0.900 - 0.435i$
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 + (−2.62 + 0.358i)7-s + (−0.707 + 1.22i)11-s + 3.24·13-s + (−1.29 + 2.23i)17-s + (2.91 + 5.04i)19-s + (−3.53 − 6.12i)23-s + (−0.499 + 0.866i)25-s − 5.07·29-s + (2.5 − 4.33i)31-s + (−1.62 − 2.09i)35-s + (5.20 + 9.01i)37-s − 0.242·41-s − 10.8·43-s + (−3.82 − 6.63i)47-s + ⋯
L(s)  = 1  + (0.223 + 0.387i)5-s + (−0.990 + 0.135i)7-s + (−0.213 + 0.369i)11-s + 0.899·13-s + (−0.313 + 0.543i)17-s + (0.668 + 1.15i)19-s + (−0.737 − 1.27i)23-s + (−0.0999 + 0.173i)25-s − 0.941·29-s + (0.449 − 0.777i)31-s + (−0.274 − 0.353i)35-s + (0.856 + 1.48i)37-s − 0.0378·41-s − 1.66·43-s + (−0.558 − 0.967i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.900 - 0.435i$
Analytic conductor: \(20.1223\)
Root analytic conductor: \(4.48578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :1/2),\ -0.900 - 0.435i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6862014624\)
\(L(\frac12)\) \(\approx\) \(0.6862014624\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.62 - 0.358i)T \)
good11 \( 1 + (0.707 - 1.22i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.24T + 13T^{2} \)
17 \( 1 + (1.29 - 2.23i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.91 - 5.04i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.53 + 6.12i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.07T + 29T^{2} \)
31 \( 1 + (-2.5 + 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.20 - 9.01i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.242T + 41T^{2} \)
43 \( 1 + 10.8T + 43T^{2} \)
47 \( 1 + (3.82 + 6.63i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.41 - 5.91i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.87 - 4.98i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.58 + 7.94i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.44 - 7.70i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.0T + 71T^{2} \)
73 \( 1 + (0.621 - 1.07i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.91 - 10.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.24T + 83T^{2} \)
89 \( 1 + (-1.12 - 1.94i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 8.48T + 97T^{2} \)
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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

−9.344505823924442102204311490965, −8.398473285213344496789063004677, −7.81906823480282205256996929949, −6.69906219897855699237324477105, −6.25525977488135840969301118744, −5.56244601373896058760371836388, −4.32319278695896215269948533264, −3.54189525779955269545735418466, −2.67108572032566234350180176525, −1.53346838680474575849061627583, 0.22305826731680300286739887253, 1.55242475397785126492197483542, 2.91539185082721658010396145417, 3.57325523646295535653475399421, 4.63407412061797132719257473271, 5.57408087840079610659985618158, 6.18774319046097782298515512902, 7.04738948549882990131822387095, 7.77994227627093904828575746979, 8.755632208924224991627111536485

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