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Properties

Label 2-2520-7.2-c1-0-15
Degree $2$
Conductor $2520$
Sign $0.605 - 0.795i$
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 + (2.5 − 0.866i)7-s + (1 + 1.73i)11-s + 5·13-s + (2 + 3.46i)17-s + (−1.5 + 2.59i)19-s + (−2 + 3.46i)23-s + (−0.499 − 0.866i)25-s − 4·29-s + (0.5 + 0.866i)31-s + (−0.500 + 2.59i)35-s + (0.5 − 0.866i)37-s − 6·41-s − 43-s + (5 − 8.66i)47-s + ⋯
L(s)  = 1  + (−0.223 + 0.387i)5-s + (0.944 − 0.327i)7-s + (0.301 + 0.522i)11-s + 1.38·13-s + (0.485 + 0.840i)17-s + (−0.344 + 0.596i)19-s + (−0.417 + 0.722i)23-s + (−0.0999 − 0.173i)25-s − 0.742·29-s + (0.0898 + 0.155i)31-s + (−0.0845 + 0.439i)35-s + (0.0821 − 0.142i)37-s − 0.937·41-s − 0.152·43-s + (0.729 − 1.26i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.605 - 0.795i$
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: \((2,\ 2520,\ (\ :1/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.034498380\)
\(L(\frac12)\) \(\approx\) \(2.034498380\)
\(L(\frac{3}{2})\) 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 \( 1 - 5T + 13T^{2} \)
17 \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.5 - 2.59i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + (-5 + 8.66i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3 + 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.5 - 12.9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (-1.5 - 2.59i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.5 - 7.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
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

−8.795201482912473612700546105280, −8.258743120895068155107739765563, −7.57697760880042947896324168853, −6.76661022998054266569106021937, −5.92132092489142434371408885566, −5.15676423351770584615404448627, −3.86813567945296121499631468035, −3.74410919493692531406211698256, −2.08388998352591601357018518138, −1.26445358578610905928520409279, 0.77009399449699687942593685180, 1.84499999121165171339269543376, 3.07499212493005001598652338628, 4.05270387075090431370973319214, 4.83216011323019532310344518042, 5.67484172234609576060893685989, 6.38105837694974546616720022035, 7.40810185472724006201442257667, 8.195564172631457662970888699361, 8.707202851721284685348416241559

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