Properties

Label 2-60e2-300.23-c0-0-0
Degree $2$
Conductor $3600$
Sign $0.470 - 0.882i$
Analytic cond. $1.79663$
Root an. cond. $1.34038$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.891 + 0.453i)5-s + (0.142 + 0.896i)13-s + (−0.734 + 1.44i)17-s + (0.587 + 0.809i)25-s + (−0.0966 + 0.297i)29-s + (−0.309 + 0.0489i)37-s + (−0.533 + 0.734i)41-s i·49-s + (0.280 + 0.550i)53-s + (1.53 − 1.11i)61-s + (−0.280 + 0.863i)65-s + (1.76 + 0.278i)73-s + (−1.30 + 0.951i)85-s + (−1.59 + 1.16i)89-s + (−0.896 − 1.76i)97-s + ⋯
L(s)  = 1  + (0.891 + 0.453i)5-s + (0.142 + 0.896i)13-s + (−0.734 + 1.44i)17-s + (0.587 + 0.809i)25-s + (−0.0966 + 0.297i)29-s + (−0.309 + 0.0489i)37-s + (−0.533 + 0.734i)41-s i·49-s + (0.280 + 0.550i)53-s + (1.53 − 1.11i)61-s + (−0.280 + 0.863i)65-s + (1.76 + 0.278i)73-s + (−1.30 + 0.951i)85-s + (−1.59 + 1.16i)89-s + (−0.896 − 1.76i)97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.470 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.470 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.470 - 0.882i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (3023, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :0),\ 0.470 - 0.882i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.426323521\)
\(L(\frac12)\) \(\approx\) \(1.426323521\)
\(L(1)\) 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 \( 1 + (-0.891 - 0.453i)T \)
good7 \( 1 + iT^{2} \)
11 \( 1 + (0.309 - 0.951i)T^{2} \)
13 \( 1 + (-0.142 - 0.896i)T + (-0.951 + 0.309i)T^{2} \)
17 \( 1 + (0.734 - 1.44i)T + (-0.587 - 0.809i)T^{2} \)
19 \( 1 + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + (0.951 + 0.309i)T^{2} \)
29 \( 1 + (0.0966 - 0.297i)T + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.309 - 0.0489i)T + (0.951 - 0.309i)T^{2} \)
41 \( 1 + (0.533 - 0.734i)T + (-0.309 - 0.951i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (-0.587 + 0.809i)T^{2} \)
53 \( 1 + (-0.280 - 0.550i)T + (-0.587 + 0.809i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2} \)
67 \( 1 + (-0.587 - 0.809i)T^{2} \)
71 \( 1 + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (-1.76 - 0.278i)T + (0.951 + 0.309i)T^{2} \)
79 \( 1 + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (-0.587 - 0.809i)T^{2} \)
89 \( 1 + (1.59 - 1.16i)T + (0.309 - 0.951i)T^{2} \)
97 \( 1 + (0.896 + 1.76i)T + (-0.587 + 0.809i)T^{2} \)
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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

−8.748371980343627743698697385259, −8.352179755662939315155535000338, −7.14226005033902694244753120153, −6.61997355140169125346538855463, −5.99421700569432475280145222617, −5.16596030562869434501174238335, −4.21579471916684780342161448997, −3.38425102285641177910995326550, −2.24611879573413677463822039234, −1.58374008495254763340833843470, 0.840549776385775673521892692054, 2.13895075802270648919704488491, 2.88514149122414953238037386621, 4.02258830871040513746725860113, 5.04587676157798213081363027539, 5.43788009067373811705047696565, 6.36893170397519025973321325506, 7.05620237490325613052502035677, 7.927669684265896303840742269391, 8.725203491430898459150574364922

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