Properties

Label 2-3552-888.221-c0-0-8
Degree $2$
Conductor $3552$
Sign $1$
Analytic cond. $1.77267$
Root an. cond. $1.33141$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 7-s + 9-s − 11-s + 13-s − 17-s − 19-s + 21-s + 23-s + 25-s + 27-s − 33-s − 37-s + 39-s + 2·43-s − 51-s + 53-s − 57-s − 2·61-s + 63-s + 69-s − 73-s + 75-s − 77-s + 81-s − 83-s − 89-s + ⋯
L(s)  = 1  + 3-s + 7-s + 9-s − 11-s + 13-s − 17-s − 19-s + 21-s + 23-s + 25-s + 27-s − 33-s − 37-s + 39-s + 2·43-s − 51-s + 53-s − 57-s − 2·61-s + 63-s + 69-s − 73-s + 75-s − 77-s + 81-s − 83-s − 89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3552\)    =    \(2^{5} \cdot 3 \cdot 37\)
Sign: $1$
Analytic conductor: \(1.77267\)
Root analytic conductor: \(1.33141\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3552} (1553, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3552,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.034756259\)
\(L(\frac12)\) \(\approx\) \(2.034756259\)
\(L(1)\) 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 \( 1 - T \)
37 \( 1 + T \)
good5 \( ( 1 - T )( 1 + T ) \)
7 \( 1 - T + T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 - T + T^{2} \)
17 \( 1 + T + T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 - T + T^{2} \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 - T )^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( 1 - T + T^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 + T )^{2} \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 + T + T^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( 1 + T + T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−8.742568584045194631529784977167, −8.164429038338974845615104962669, −7.39062442445108354912202056053, −6.73246939899289482739192092015, −5.68335419139757408066672874561, −4.70746932313453597968315584941, −4.21386640679277210643173533016, −3.08035179140500473160089484584, −2.31985145558842027265647597899, −1.35731215397276712332445410110, 1.35731215397276712332445410110, 2.31985145558842027265647597899, 3.08035179140500473160089484584, 4.21386640679277210643173533016, 4.70746932313453597968315584941, 5.68335419139757408066672874561, 6.73246939899289482739192092015, 7.39062442445108354912202056053, 8.164429038338974845615104962669, 8.742568584045194631529784977167

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