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Properties

Label 2-60e2-300.203-c0-0-1
Degree $2$
Conductor $3600$
Sign $-0.0755 + 0.997i$
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  + (0.156 − 0.987i)5-s + (0.896 − 1.76i)13-s + (−0.610 − 0.0966i)17-s + (−0.951 − 0.309i)25-s + (−0.734 − 0.533i)29-s + (0.809 + 0.412i)37-s + (−1.87 + 0.610i)41-s − i·49-s + (1.59 − 0.253i)53-s + (−0.363 + 1.11i)61-s + (−1.59 − 1.16i)65-s + (0.278 − 0.142i)73-s + (−0.190 + 0.587i)85-s + (0.550 − 1.69i)89-s + (1.76 − 0.278i)97-s + ⋯
L(s)  = 1  + (0.156 − 0.987i)5-s + (0.896 − 1.76i)13-s + (−0.610 − 0.0966i)17-s + (−0.951 − 0.309i)25-s + (−0.734 − 0.533i)29-s + (0.809 + 0.412i)37-s + (−1.87 + 0.610i)41-s − i·49-s + (1.59 − 0.253i)53-s + (−0.363 + 1.11i)61-s + (−1.59 − 1.16i)65-s + (0.278 − 0.142i)73-s + (−0.190 + 0.587i)85-s + (0.550 − 1.69i)89-s + (1.76 − 0.278i)97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0755 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0755 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.0755 + 0.997i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (2303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :0),\ -0.0755 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.196724170\)
\(L(\frac12)\) \(\approx\) \(1.196724170\)
\(L(1)\) 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 \( 1 + (-0.156 + 0.987i)T \)
good7 \( 1 + iT^{2} \)
11 \( 1 + (-0.809 - 0.587i)T^{2} \)
13 \( 1 + (-0.896 + 1.76i)T + (-0.587 - 0.809i)T^{2} \)
17 \( 1 + (0.610 + 0.0966i)T + (0.951 + 0.309i)T^{2} \)
19 \( 1 + (0.309 - 0.951i)T^{2} \)
23 \( 1 + (0.587 - 0.809i)T^{2} \)
29 \( 1 + (0.734 + 0.533i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.809 - 0.412i)T + (0.587 + 0.809i)T^{2} \)
41 \( 1 + (1.87 - 0.610i)T + (0.809 - 0.587i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (0.951 - 0.309i)T^{2} \)
53 \( 1 + (-1.59 + 0.253i)T + (0.951 - 0.309i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.363 - 1.11i)T + (-0.809 - 0.587i)T^{2} \)
67 \( 1 + (0.951 + 0.309i)T^{2} \)
71 \( 1 + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.278 + 0.142i)T + (0.587 - 0.809i)T^{2} \)
79 \( 1 + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.951 + 0.309i)T^{2} \)
89 \( 1 + (-0.550 + 1.69i)T + (-0.809 - 0.587i)T^{2} \)
97 \( 1 + (-1.76 + 0.278i)T + (0.951 - 0.309i)T^{2} \)
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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

−8.437158815216965625003277849061, −8.062231833181424253803742489624, −7.12179458604650262776765993068, −6.08978635254481733153249457871, −5.55976092386691760361522151260, −4.81310247356433936247705298518, −3.91106776175062514125337173662, −3.03754286727657892250559330378, −1.84821757138007885464556408404, −0.69900473873621712118897565285, 1.64145523951003654045980133972, 2.41947158930724095630113378281, 3.58551380680103488750504367361, 4.11580368540707788380055640994, 5.19732219769330681461946721530, 6.19224062830832542149036722516, 6.66529779717285786799880094166, 7.26744890328745977725775672200, 8.192251330977358747064160232755, 9.079958101656393563536377603060

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