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Properties

Label 2-3648-456.149-c0-0-6
Degree $2$
Conductor $3648$
Sign $0.102 + 0.994i$
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 + (0.342 + 0.592i)7-s + (−0.766 + 0.642i)9-s + (0.673 − 1.85i)13-s + (−0.866 + 0.5i)19-s + (0.439 − 0.524i)21-s + (0.939 + 0.342i)25-s + (0.866 + 0.500i)27-s + (−0.642 − 1.11i)31-s − 1.28i·37-s − 1.96·39-s + (1.50 + 0.266i)43-s + (0.266 − 0.460i)49-s + (0.766 + 0.642i)57-s + (1.93 − 0.342i)61-s + ⋯
L(s)  = 1  + (−0.342 − 0.939i)3-s + (0.342 + 0.592i)7-s + (−0.766 + 0.642i)9-s + (0.673 − 1.85i)13-s + (−0.866 + 0.5i)19-s + (0.439 − 0.524i)21-s + (0.939 + 0.342i)25-s + (0.866 + 0.500i)27-s + (−0.642 − 1.11i)31-s − 1.28i·37-s − 1.96·39-s + (1.50 + 0.266i)43-s + (0.266 − 0.460i)49-s + (0.766 + 0.642i)57-s + (1.93 − 0.342i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.102 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.102 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3648\)    =    \(2^{6} \cdot 3 \cdot 19\)
Sign: $0.102 + 0.994i$
Analytic conductor: \(1.82058\)
Root analytic conductor: \(1.34929\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3648} (2657, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3648,\ (\ :0),\ 0.102 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.104709745\)
\(L(\frac12)\) \(\approx\) \(1.104709745\)
\(L(1)\) 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 \( 1 + (-0.342 - 0.592i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.673 + 1.85i)T + (-0.766 - 0.642i)T^{2} \)
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 \( 1 + 1.28iT - T^{2} \)
41 \( 1 + (-0.766 + 0.642i)T^{2} \)
43 \( 1 + (-1.50 - 0.266i)T + (0.939 + 0.342i)T^{2} \)
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 \( 1 + (-0.766 - 0.642i)T + (0.173 + 0.984i)T^{2} \)
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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

−8.437289655278272885734924418190, −7.79029674545392044928433808225, −7.19750649212824169611586281410, −6.07958846245398796616187100183, −5.76674126710643790007514189743, −5.03875291466879655011082808881, −3.80900697692555575963863437216, −2.78843179995964902527809595043, −1.97528642402283118493391809573, −0.74691215163744879068571678777, 1.27023005822810214205594838684, 2.57651159838330299900592779598, 3.73157495065564976124787930276, 4.35055135426310951433783789538, 4.85613866596244624238947412149, 5.91795735275581209781406327768, 6.66277094135110909291632553774, 7.20785178416174543917893756700, 8.552598125444649036323477447654, 8.804292909765884649859417836247

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