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Properties

Label 2-3648-456.155-c0-0-2
Degree $2$
Conductor $3648$
Sign $0.706 - 0.707i$
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  + (0.342 + 0.939i)3-s + (1.62 − 0.939i)7-s + (−0.766 + 0.642i)9-s + (−0.326 − 0.118i)13-s + (−0.866 + 0.5i)19-s + (1.43 + 1.20i)21-s + (0.939 + 0.342i)25-s + (−0.866 − 0.500i)27-s + (0.642 + 1.11i)31-s + 1.53·37-s − 0.347i·39-s + (−0.223 + 1.26i)43-s + (1.26 − 2.19i)49-s + (−0.766 − 0.642i)57-s + (1.93 − 0.342i)61-s + ⋯
L(s)  = 1  + (0.342 + 0.939i)3-s + (1.62 − 0.939i)7-s + (−0.766 + 0.642i)9-s + (−0.326 − 0.118i)13-s + (−0.866 + 0.5i)19-s + (1.43 + 1.20i)21-s + (0.939 + 0.342i)25-s + (−0.866 − 0.500i)27-s + (0.642 + 1.11i)31-s + 1.53·37-s − 0.347i·39-s + (−0.223 + 1.26i)43-s + (1.26 − 2.19i)49-s + (−0.766 − 0.642i)57-s + (1.93 − 0.342i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.706 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.706 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3648\)    =    \(2^{6} \cdot 3 \cdot 19\)
Sign: $0.706 - 0.707i$
Analytic conductor: \(1.82058\)
Root analytic conductor: \(1.34929\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3648} (2207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3648,\ (\ :0),\ 0.706 - 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.682223184\)
\(L(\frac12)\) \(\approx\) \(1.682223184\)
\(L(1)\) 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 \( 1 + (-0.342 - 0.939i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
good5 \( 1 + (-0.939 - 0.342i)T^{2} \)
7 \( 1 + (-1.62 + 0.939i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \)
17 \( 1 + (-0.173 - 0.984i)T^{2} \)
23 \( 1 + (-0.939 + 0.342i)T^{2} \)
29 \( 1 + (-0.173 + 0.984i)T^{2} \)
31 \( 1 + (-0.642 - 1.11i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - 1.53T + T^{2} \)
41 \( 1 + (0.766 - 0.642i)T^{2} \)
43 \( 1 + (0.223 - 1.26i)T + (-0.939 - 0.342i)T^{2} \)
47 \( 1 + (0.173 - 0.984i)T^{2} \)
53 \( 1 + (0.939 - 0.342i)T^{2} \)
59 \( 1 + (0.173 + 0.984i)T^{2} \)
61 \( 1 + (-1.93 + 0.342i)T + (0.939 - 0.342i)T^{2} \)
67 \( 1 + (1.20 + 1.43i)T + (-0.173 + 0.984i)T^{2} \)
71 \( 1 + (0.939 + 0.342i)T^{2} \)
73 \( 1 + (-0.326 + 0.118i)T + (0.766 - 0.642i)T^{2} \)
79 \( 1 + (0.642 - 0.233i)T + (0.766 - 0.642i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.766 + 0.642i)T^{2} \)
97 \( 1 + (1.11 - 1.32i)T + (-0.173 - 0.984i)T^{2} \)
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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

−8.626182778939964441840473131870, −8.187438478758038006119305243991, −7.58672954570832802048422762301, −6.65686051345384031843826510113, −5.56450320684900908865916124224, −4.69689171035528813995339111655, −4.46306888172159164594102091683, −3.47865531777838368980097476160, −2.43296300720504804325540080300, −1.31097031921841148198004553447, 1.12384331894468517352029343685, 2.28062067432596912565270717313, 2.56921897699694120144428829184, 4.09369788534276274374593489510, 4.89958782692211512271004307908, 5.68975510391435525889237314017, 6.42685582535670168073247181789, 7.31050019160207293911363864447, 7.929847106481507552006537490176, 8.635394146257027006338479109095

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