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Properties

Label 2-3660-3660.1619-c0-0-6
Degree $2$
Conductor $3660$
Sign $0.724 + 0.689i$
Analytic cond. $1.82657$
Root an. cond. $1.35150$
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  + (0.309 + 0.951i)2-s + (0.309 − 0.951i)3-s + (−0.809 + 0.587i)4-s + (0.587 − 0.809i)5-s + 0.999·6-s + (−0.809 − 0.587i)8-s + (−0.809 − 0.587i)9-s + (0.951 + 0.309i)10-s + (0.309 + 0.951i)12-s + (−0.587 − 0.809i)15-s + (0.309 − 0.951i)16-s + (0.221 + 1.39i)17-s + (0.309 − 0.951i)18-s + (0.587 − 1.80i)19-s + 0.999i·20-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + (0.309 − 0.951i)3-s + (−0.809 + 0.587i)4-s + (0.587 − 0.809i)5-s + 0.999·6-s + (−0.809 − 0.587i)8-s + (−0.809 − 0.587i)9-s + (0.951 + 0.309i)10-s + (0.309 + 0.951i)12-s + (−0.587 − 0.809i)15-s + (0.309 − 0.951i)16-s + (0.221 + 1.39i)17-s + (0.309 − 0.951i)18-s + (0.587 − 1.80i)19-s + 0.999i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.724 + 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.724 + 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3660\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 61\)
Sign: $0.724 + 0.689i$
Analytic conductor: \(1.82657\)
Root analytic conductor: \(1.35150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3660} (1619, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3660,\ (\ :0),\ 0.724 + 0.689i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.521537228\)
\(L(\frac12)\) \(\approx\) \(1.521537228\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
3 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + (-0.587 + 0.809i)T \)
61 \( 1 + (-0.587 + 0.809i)T \)
good7 \( 1 + (0.587 + 0.809i)T^{2} \)
11 \( 1 + iT^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (-0.221 - 1.39i)T + (-0.951 + 0.309i)T^{2} \)
19 \( 1 + (-0.587 + 1.80i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + (0.142 + 0.896i)T + (-0.951 + 0.309i)T^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 + (-0.278 - 0.142i)T + (0.587 + 0.809i)T^{2} \)
37 \( 1 + (0.587 + 0.809i)T^{2} \)
41 \( 1 + (-0.809 + 0.587i)T^{2} \)
43 \( 1 + (0.951 + 0.309i)T^{2} \)
47 \( 1 + 2iT - T^{2} \)
53 \( 1 + (0.309 - 1.95i)T + (-0.951 - 0.309i)T^{2} \)
59 \( 1 + (0.587 - 0.809i)T^{2} \)
67 \( 1 + (-0.951 + 0.309i)T^{2} \)
71 \( 1 + (0.951 + 0.309i)T^{2} \)
73 \( 1 + (-0.309 + 0.951i)T^{2} \)
79 \( 1 + (-1.76 - 0.278i)T + (0.951 + 0.309i)T^{2} \)
83 \( 1 + (1.11 + 0.363i)T + (0.809 + 0.587i)T^{2} \)
89 \( 1 + (-0.587 + 0.809i)T^{2} \)
97 \( 1 + (-0.809 + 0.587i)T^{2} \)
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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

−8.449632959291953224306458599118, −7.969873056976194416082569892526, −6.97620583157797870316996174631, −6.51343698664639402301992902137, −5.73388809507904779409462439193, −5.07316645592561273910050872700, −4.19289134526335491557192310758, −3.12623147945892847303020787440, −2.08721804367566109681971664054, −0.78496527257163316510997177666, 1.58185581029134552317334601405, 2.64710367730872582352376753735, 3.26058726329384025474244537768, 3.93503453650663346614358492370, 4.96607640785476144315092049494, 5.57143467936764735866387589094, 6.24507424768825204877035825520, 7.50553596558390666613179213054, 8.205459637866006793097278293740, 9.278239287051131795687247433663

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