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Properties

Label 2-2520-7.4-c1-0-19
Degree $2$
Conductor $2520$
Sign $0.491 - 0.871i$
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 + (0.194 + 2.63i)7-s + (−1.26 + 2.19i)11-s + 5.45·13-s + (3.25 − 5.64i)17-s + (−0.0406 − 0.0703i)19-s + (−1.23 − 2.13i)23-s + (−0.499 + 0.866i)25-s + 8.37·29-s + (−0.852 + 1.47i)31-s + (−2.18 + 1.48i)35-s + (1.18 + 2.05i)37-s − 3.75·41-s + 9.44·43-s + (0.962 + 1.66i)47-s + ⋯
L(s)  = 1  + (0.223 + 0.387i)5-s + (0.0736 + 0.997i)7-s + (−0.381 + 0.660i)11-s + 1.51·13-s + (0.790 − 1.36i)17-s + (−0.00932 − 0.0161i)19-s + (−0.257 − 0.446i)23-s + (−0.0999 + 0.173i)25-s + 1.55·29-s + (−0.153 + 0.265i)31-s + (−0.369 + 0.251i)35-s + (0.195 + 0.338i)37-s − 0.585·41-s + 1.43·43-s + (0.140 + 0.243i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.491 - 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.491 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.491 - 0.871i$
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: \((2,\ 2520,\ (\ :1/2),\ 0.491 - 0.871i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.033278106\)
\(L(\frac12)\) \(\approx\) \(2.033278106\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.194 - 2.63i)T \)
good11 \( 1 + (1.26 - 2.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.45T + 13T^{2} \)
17 \( 1 + (-3.25 + 5.64i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0406 + 0.0703i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.23 + 2.13i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.37T + 29T^{2} \)
31 \( 1 + (0.852 - 1.47i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.18 - 2.05i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.75T + 41T^{2} \)
43 \( 1 - 9.44T + 43T^{2} \)
47 \( 1 + (-0.962 - 1.66i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.49 - 4.31i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.65 - 2.87i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.150 + 0.261i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.58 - 2.74i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.684T + 71T^{2} \)
73 \( 1 + (7.64 - 13.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.29 - 12.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12.0T + 83T^{2} \)
89 \( 1 + (1.79 + 3.11i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.20T + 97T^{2} \)
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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

−8.983219408740244363816354595045, −8.365410574563844076157722867745, −7.53313580337458339794097773256, −6.65713047361655898975273839990, −5.93819822024455989528056482941, −5.22664364116871466359738562573, −4.31664668057580995128712474505, −3.09197132159516156917120414608, −2.48560467447510033594333462652, −1.19263882247006801524469833321, 0.802296513911748944531364640476, 1.68769441399858056452765235851, 3.25261926338178642609988817756, 3.86432234936375746489233128126, 4.76335486739982228089218686995, 5.92011119964527599615856280668, 6.20072255443770818690651357911, 7.37270125700288148716145846259, 8.176972672237972523262981522527, 8.559301109815302539655441807778

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