Properties

Label 2-3267-297.119-c0-0-0
Degree $2$
Conductor $3267$
Sign $-0.491 - 0.871i$
Analytic cond. $1.63044$
Root an. cond. $1.27688$
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  + (0.241 − 0.970i)3-s + (0.438 − 0.898i)4-s + (−1.96 − 0.0687i)5-s + (−0.882 − 0.469i)9-s + (−0.766 − 0.642i)12-s + (−0.542 + 1.89i)15-s + (−0.615 − 0.788i)16-s + (−0.924 + 1.73i)20-s + (−1.70 + 0.300i)23-s + (2.87 + 0.200i)25-s + (−0.669 + 0.743i)27-s + (−1.51 − 0.213i)31-s + (−0.809 + 0.587i)36-s + (0.0363 − 0.345i)37-s + (1.70 + 0.984i)45-s + ⋯
L(s)  = 1  + (0.241 − 0.970i)3-s + (0.438 − 0.898i)4-s + (−1.96 − 0.0687i)5-s + (−0.882 − 0.469i)9-s + (−0.766 − 0.642i)12-s + (−0.542 + 1.89i)15-s + (−0.615 − 0.788i)16-s + (−0.924 + 1.73i)20-s + (−1.70 + 0.300i)23-s + (2.87 + 0.200i)25-s + (−0.669 + 0.743i)27-s + (−1.51 − 0.213i)31-s + (−0.809 + 0.587i)36-s + (0.0363 − 0.345i)37-s + (1.70 + 0.984i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.491 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.491 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3267\)    =    \(3^{3} \cdot 11^{2}\)
Sign: $-0.491 - 0.871i$
Analytic conductor: \(1.63044\)
Root analytic conductor: \(1.27688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3267} (2792, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3267,\ (\ :0),\ -0.491 - 0.871i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2758085399\)
\(L(\frac12)\) \(\approx\) \(0.2758085399\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.241 + 0.970i)T \)
11 \( 1 \)
good2 \( 1 + (-0.438 + 0.898i)T^{2} \)
5 \( 1 + (1.96 + 0.0687i)T + (0.997 + 0.0697i)T^{2} \)
7 \( 1 + (0.0348 - 0.999i)T^{2} \)
13 \( 1 + (-0.719 + 0.694i)T^{2} \)
17 \( 1 + (0.104 + 0.994i)T^{2} \)
19 \( 1 + (-0.978 + 0.207i)T^{2} \)
23 \( 1 + (1.70 - 0.300i)T + (0.939 - 0.342i)T^{2} \)
29 \( 1 + (0.882 - 0.469i)T^{2} \)
31 \( 1 + (1.51 + 0.213i)T + (0.961 + 0.275i)T^{2} \)
37 \( 1 + (-0.0363 + 0.345i)T + (-0.978 - 0.207i)T^{2} \)
41 \( 1 + (0.882 + 0.469i)T^{2} \)
43 \( 1 + (0.766 - 0.642i)T^{2} \)
47 \( 1 + (0.614 - 0.299i)T + (0.615 - 0.788i)T^{2} \)
53 \( 1 + (-1.22 - 0.397i)T + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (-1.06 + 0.718i)T + (0.374 - 0.927i)T^{2} \)
61 \( 1 + (0.961 - 0.275i)T^{2} \)
67 \( 1 + (1.17 - 0.984i)T + (0.173 - 0.984i)T^{2} \)
71 \( 1 + (-0.508 - 0.457i)T + (0.104 + 0.994i)T^{2} \)
73 \( 1 + (0.669 - 0.743i)T^{2} \)
79 \( 1 + (0.438 - 0.898i)T^{2} \)
83 \( 1 + (0.719 + 0.694i)T^{2} \)
89 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.0655 + 1.87i)T + (-0.997 + 0.0697i)T^{2} \)
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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

−8.197906420535319666694901939006, −7.39590029307197950919730838659, −7.16618536868887022360716075060, −6.16770479339924260770515586204, −5.43820687010410431390736960341, −4.31040202153039419753588723407, −3.56628863914907215023739515600, −2.53793004300170031688407726671, −1.41547899508825212609182599351, −0.15806215625776745071411809700, 2.31684592081331610496428044213, 3.35383730531335035300633659935, 3.81162484224979743829430265034, 4.32547659975166832220864256477, 5.29589344382228591030994932870, 6.54019061207758440162275531246, 7.35394719184436487382024757182, 7.958484685278723874152546622203, 8.427293423619262093139259463748, 9.032827693254068163363318089239

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