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Properties

Label 2-3648-456.365-c0-0-6
Degree $2$
Conductor $3648$
Sign $0.796 + 0.605i$
Analytic cond. $1.82058$
Root an. cond. $1.34929$
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  + (0.984 + 0.173i)3-s + (0.939 + 0.342i)9-s + (−0.984 − 1.70i)11-s + (0.592 − 1.62i)17-s + (0.642 − 0.766i)19-s + (−0.173 + 0.984i)25-s + (0.866 + 0.5i)27-s + (−0.673 − 1.85i)33-s + (−1.26 + 0.223i)41-s + (−0.642 − 0.766i)43-s + (0.5 + 0.866i)49-s + (0.866 − 1.5i)51-s + (0.766 − 0.642i)57-s + (1.85 + 0.673i)59-s + (−0.524 − 1.43i)67-s + ⋯
L(s)  = 1  + (0.984 + 0.173i)3-s + (0.939 + 0.342i)9-s + (−0.984 − 1.70i)11-s + (0.592 − 1.62i)17-s + (0.642 − 0.766i)19-s + (−0.173 + 0.984i)25-s + (0.866 + 0.5i)27-s + (−0.673 − 1.85i)33-s + (−1.26 + 0.223i)41-s + (−0.642 − 0.766i)43-s + (0.5 + 0.866i)49-s + (0.866 − 1.5i)51-s + (0.766 − 0.642i)57-s + (1.85 + 0.673i)59-s + (−0.524 − 1.43i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.796 + 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.796 + 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3648\)    =    \(2^{6} \cdot 3 \cdot 19\)
Sign: $0.796 + 0.605i$
Analytic conductor: \(1.82058\)
Root analytic conductor: \(1.34929\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3648} (1505, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3648,\ (\ :0),\ 0.796 + 0.605i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.771688259\)
\(L(\frac12)\) \(\approx\) \(1.771688259\)
\(L(1)\) 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 \( 1 + (-0.984 - 0.173i)T \)
19 \( 1 + (-0.642 + 0.766i)T \)
good5 \( 1 + (0.173 - 0.984i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.984 + 1.70i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.939 + 0.342i)T^{2} \)
17 \( 1 + (-0.592 + 1.62i)T + (-0.766 - 0.642i)T^{2} \)
23 \( 1 + (-0.173 - 0.984i)T^{2} \)
29 \( 1 + (0.766 - 0.642i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (1.26 - 0.223i)T + (0.939 - 0.342i)T^{2} \)
43 \( 1 + (0.642 + 0.766i)T + (-0.173 + 0.984i)T^{2} \)
47 \( 1 + (-0.766 + 0.642i)T^{2} \)
53 \( 1 + (0.173 + 0.984i)T^{2} \)
59 \( 1 + (-1.85 - 0.673i)T + (0.766 + 0.642i)T^{2} \)
61 \( 1 + (-0.173 - 0.984i)T^{2} \)
67 \( 1 + (0.524 + 1.43i)T + (-0.766 + 0.642i)T^{2} \)
71 \( 1 + (-0.173 + 0.984i)T^{2} \)
73 \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \)
79 \( 1 + (-0.939 + 0.342i)T^{2} \)
83 \( 1 + (0.642 - 1.11i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (-1.70 - 0.300i)T + (0.939 + 0.342i)T^{2} \)
97 \( 1 + (-1.43 - 0.524i)T + (0.766 + 0.642i)T^{2} \)
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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

−8.762010809862934916659358327807, −7.85108638894539889829737042022, −7.45198972553148557193512616093, −6.55510365160858595890345541378, −5.30434768600568420108627641660, −5.08852501163917657357948832792, −3.68279746332861527512513371104, −3.11069637083326590936075746099, −2.45750924604200177437153775365, −0.953782183387318593115696336111, 1.60565034033900899601810210616, 2.23230158869384251179304366468, 3.30157055361918722670584628349, 4.08026591731518938181641787379, 4.87987001330719608855420080461, 5.82324040755104973501310443369, 6.81380425985596889473925895333, 7.43864280397100319588822310277, 8.140805700756887183400755094462, 8.517213896324654279251070930488

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