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Properties

Label 2-2520-7.4-c1-0-29
Degree $2$
Conductor $2520$
Sign $0.845 + 0.533i$
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 + (2.64 − 0.0641i)7-s + (2.91 − 5.04i)11-s + 2.75·13-s + (1 − 1.73i)17-s + (0.378 + 0.654i)19-s + (−0.266 − 0.462i)23-s + (−0.499 + 0.866i)25-s + 0.823·29-s + (1.28 − 2.23i)31-s + (1.37 + 2.25i)35-s + (−2.37 − 4.11i)37-s − 6.06·41-s + 0.710·43-s + (−6.44 − 11.1i)47-s + ⋯
L(s)  = 1  + (0.223 + 0.387i)5-s + (0.999 − 0.0242i)7-s + (0.877 − 1.52i)11-s + 0.764·13-s + (0.242 − 0.420i)17-s + (0.0867 + 0.150i)19-s + (−0.0556 − 0.0963i)23-s + (−0.0999 + 0.173i)25-s + 0.152·29-s + (0.231 − 0.401i)31-s + (0.232 + 0.381i)35-s + (−0.390 − 0.677i)37-s − 0.947·41-s + 0.108·43-s + (−0.940 − 1.62i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 + 0.533i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.845 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.845 + 0.533i$
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: \((2,\ 2520,\ (\ :1/2),\ 0.845 + 0.533i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.345021154\)
\(L(\frac12)\) \(\approx\) \(2.345021154\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.64 + 0.0641i)T \)
good11 \( 1 + (-2.91 + 5.04i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.75T + 13T^{2} \)
17 \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.378 - 0.654i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.266 + 0.462i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.823T + 29T^{2} \)
31 \( 1 + (-1.28 + 2.23i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.37 + 4.11i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.06T + 41T^{2} \)
43 \( 1 - 0.710T + 43T^{2} \)
47 \( 1 + (6.44 + 11.1i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.20 - 7.27i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4 + 6.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.70 + 8.14i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.93 - 10.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (1.75 - 3.04i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.75 - 8.23i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.71T + 83T^{2} \)
89 \( 1 + (-0.878 - 1.52i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−8.635701653343796622229391527348, −8.297977943482403534777774283069, −7.31852616629669256257083249223, −6.44074148832402404430416455573, −5.80261972744394702155787776020, −4.98917532315911908998536785702, −3.87157526686332109610631984902, −3.21978733736977085283752556097, −1.93333930296574422072522946703, −0.884993035979951180712521791554, 1.34241231846071309333346306651, 1.88320036207281561775445053153, 3.33462523463145888010073948593, 4.45257862296123434899973296752, 4.81217036152832827762070873599, 5.89095273838419726027990616394, 6.68350625812677753748050836481, 7.50704210151835759774146721272, 8.274628793925968380083157365560, 8.929033283474155179930300734654

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