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Properties

Label 2-2520-7.4-c1-0-37
Degree $2$
Conductor $2520$
Sign $-0.968 - 0.250i$
Analytic cond. $20.1223$
Root an. cond. $4.48578$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)5-s + (−2.5 + 0.866i)7-s + (1 − 1.73i)11-s + (2 − 3.46i)17-s + (1 + 1.73i)19-s + (0.5 + 0.866i)23-s + (−0.499 + 0.866i)25-s − 9·29-s + (−2 + 3.46i)31-s + (2 + 1.73i)35-s + (−2 − 3.46i)37-s − 41-s + 9·43-s + (5.5 − 4.33i)49-s + (−5 + 8.66i)53-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (−0.944 + 0.327i)7-s + (0.301 − 0.522i)11-s + (0.485 − 0.840i)17-s + (0.229 + 0.397i)19-s + (0.104 + 0.180i)23-s + (−0.0999 + 0.173i)25-s − 1.67·29-s + (−0.359 + 0.622i)31-s + (0.338 + 0.292i)35-s + (−0.328 − 0.569i)37-s − 0.156·41-s + 1.37·43-s + (0.785 − 0.618i)49-s + (−0.686 + 1.18i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.968 - 0.250i$
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: \(1\)
Selberg data: \((2,\ 2520,\ (\ :1/2),\ -0.968 - 0.250i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 - 0.866i)T \)
good11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (-2 + 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 9T + 29T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + T + 41T^{2} \)
43 \( 1 - 9T + 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5 - 8.66i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5 - 8.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.5 + 7.79i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.5 - 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 14T + 71T^{2} \)
73 \( 1 + (6 - 10.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7 + 12.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 11T + 83T^{2} \)
89 \( 1 + (7.5 + 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 18T + 97T^{2} \)
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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

−8.752274737890926125497851993785, −7.55727630841637006617590457473, −7.16093231125410410239474813055, −5.90357466586885373991968913980, −5.64244382343742914093823394506, −4.43775639183383527235852529363, −3.52618566246216846742119612974, −2.81375432723879144578048216799, −1.39525055705899082171957724430, 0, 1.60643583341045968904385440330, 2.85189617137491825838976769633, 3.67890974132752309277195455038, 4.37646315367087273935995257028, 5.59907544177181933826568686564, 6.26955698911630320483055034936, 7.12597671835263987113451307540, 7.58574843349104703752427644191, 8.600156195859458894142793687162

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