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Properties

Label 2-3216-201.158-c0-0-0
Degree $2$
Conductor $3216$
Sign $-0.463 + 0.886i$
Analytic cond. $1.60499$
Root an. cond. $1.26688$
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  + (−0.841 + 0.540i)3-s + (0.544 − 1.19i)7-s + (0.415 − 0.909i)9-s + (−0.797 − 0.234i)13-s + (−0.698 − 1.53i)19-s + (0.186 + 1.29i)21-s + (−0.959 − 0.281i)25-s + (0.142 + 0.989i)27-s + (−0.273 + 0.0801i)31-s − 1.91·37-s + (0.797 − 0.234i)39-s + (−1.25 + 1.45i)43-s + (−0.468 − 0.540i)49-s + (1.41 + 0.909i)57-s + (−0.239 − 1.66i)61-s + ⋯
L(s)  = 1  + (−0.841 + 0.540i)3-s + (0.544 − 1.19i)7-s + (0.415 − 0.909i)9-s + (−0.797 − 0.234i)13-s + (−0.698 − 1.53i)19-s + (0.186 + 1.29i)21-s + (−0.959 − 0.281i)25-s + (0.142 + 0.989i)27-s + (−0.273 + 0.0801i)31-s − 1.91·37-s + (0.797 − 0.234i)39-s + (−1.25 + 1.45i)43-s + (−0.468 − 0.540i)49-s + (1.41 + 0.909i)57-s + (−0.239 − 1.66i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.463 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.463 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3216\)    =    \(2^{4} \cdot 3 \cdot 67\)
Sign: $-0.463 + 0.886i$
Analytic conductor: \(1.60499\)
Root analytic conductor: \(1.26688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3216} (2369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3216,\ (\ :0),\ -0.463 + 0.886i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5289796546\)
\(L(\frac12)\) \(\approx\) \(0.5289796546\)
\(L(1)\) 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 \( 1 + (0.841 - 0.540i)T \)
67 \( 1 + (-0.959 - 0.281i)T \)
good5 \( 1 + (0.959 + 0.281i)T^{2} \)
7 \( 1 + (-0.544 + 1.19i)T + (-0.654 - 0.755i)T^{2} \)
11 \( 1 + (0.959 + 0.281i)T^{2} \)
13 \( 1 + (0.797 + 0.234i)T + (0.841 + 0.540i)T^{2} \)
17 \( 1 + (0.142 + 0.989i)T^{2} \)
19 \( 1 + (0.698 + 1.53i)T + (-0.654 + 0.755i)T^{2} \)
23 \( 1 + (-0.415 + 0.909i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.273 - 0.0801i)T + (0.841 - 0.540i)T^{2} \)
37 \( 1 + 1.91T + T^{2} \)
41 \( 1 + (0.142 + 0.989i)T^{2} \)
43 \( 1 + (1.25 - 1.45i)T + (-0.142 - 0.989i)T^{2} \)
47 \( 1 + (-0.415 + 0.909i)T^{2} \)
53 \( 1 + (0.142 - 0.989i)T^{2} \)
59 \( 1 + (-0.841 + 0.540i)T^{2} \)
61 \( 1 + (0.239 + 1.66i)T + (-0.959 + 0.281i)T^{2} \)
71 \( 1 + (0.142 - 0.989i)T^{2} \)
73 \( 1 + (-0.0405 - 0.281i)T + (-0.959 + 0.281i)T^{2} \)
79 \( 1 + (1.84 + 0.540i)T + (0.841 + 0.540i)T^{2} \)
83 \( 1 + (0.959 + 0.281i)T^{2} \)
89 \( 1 + (-0.415 - 0.909i)T^{2} \)
97 \( 1 - 1.68T + T^{2} \)
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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

−8.637322616264086959614104007244, −7.72702612716764118367969778477, −6.98981155769894544185729458737, −6.46699177001312184749168055445, −5.32954934096106080269834046967, −4.74863134224319635961267709234, −4.14234991907198829521347885934, −3.16813709155864463561006115575, −1.74648742397330497905908348314, −0.33948603374216275500986429955, 1.74032095961211331070223656552, 2.18992389403935612858383460358, 3.63760592673497916987219729381, 4.71625390051147910156741035805, 5.47997952967927170134077541226, 5.87908681036312086617584730975, 6.83424994413530122483504164231, 7.53396581713877216343582175223, 8.329212871271315056946017739926, 8.888351268132571993043996057513

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