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Properties

Label 2-3267-99.7-c0-0-0
Degree $2$
Conductor $3267$
Sign $0.999 - 0.00829i$
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  + (−0.575 − 1.29i)2-s + (−0.669 + 0.743i)4-s + (0.913 + 0.406i)5-s + (−0.294 + 1.38i)7-s − 1.41i·10-s + (1.95 − 0.415i)14-s + (0.104 + 0.994i)16-s + (−0.913 + 0.406i)20-s + (−0.831 − 1.14i)28-s + (−0.294 + 1.38i)29-s + (−0.104 + 0.994i)31-s + (1.22 − 0.707i)32-s + (−0.831 + 1.14i)35-s + (−0.309 − 0.951i)37-s + (0.669 + 0.743i)47-s + ⋯
L(s)  = 1  + (−0.575 − 1.29i)2-s + (−0.669 + 0.743i)4-s + (0.913 + 0.406i)5-s + (−0.294 + 1.38i)7-s − 1.41i·10-s + (1.95 − 0.415i)14-s + (0.104 + 0.994i)16-s + (−0.913 + 0.406i)20-s + (−0.831 − 1.14i)28-s + (−0.294 + 1.38i)29-s + (−0.104 + 0.994i)31-s + (1.22 − 0.707i)32-s + (−0.831 + 1.14i)35-s + (−0.309 − 0.951i)37-s + (0.669 + 0.743i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3267\)    =    \(3^{3} \cdot 11^{2}\)
Sign: $0.999 - 0.00829i$
Analytic conductor: \(1.63044\)
Root analytic conductor: \(1.27688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3267} (1855, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3267,\ (\ :0),\ 0.999 - 0.00829i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8910275246\)
\(L(\frac12)\) \(\approx\) \(0.8910275246\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.575 + 1.29i)T + (-0.669 + 0.743i)T^{2} \)
5 \( 1 + (-0.913 - 0.406i)T + (0.669 + 0.743i)T^{2} \)
7 \( 1 + (0.294 - 1.38i)T + (-0.913 - 0.406i)T^{2} \)
13 \( 1 + (0.978 + 0.207i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.294 - 1.38i)T + (-0.913 - 0.406i)T^{2} \)
31 \( 1 + (0.104 - 0.994i)T + (-0.978 - 0.207i)T^{2} \)
37 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (-0.913 + 0.406i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.669 - 0.743i)T + (-0.104 + 0.994i)T^{2} \)
53 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.669 - 0.743i)T + (-0.104 - 0.994i)T^{2} \)
61 \( 1 + (1.40 - 0.147i)T + (0.978 - 0.207i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (-1.34 + 0.437i)T + (0.809 - 0.587i)T^{2} \)
79 \( 1 + (-0.669 + 0.743i)T^{2} \)
83 \( 1 + (-1.40 + 0.147i)T + (0.978 - 0.207i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-0.913 + 0.406i)T + (0.669 - 0.743i)T^{2} \)
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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

−9.089570202700606638558620053631, −8.616257537139276432941551700671, −7.44149658390938591593807875921, −6.39210033472130295445899842729, −5.86664401785938763788568586142, −5.07225826085276806396018679617, −3.71668615232628935938193661629, −2.83282263279884880722647868031, −2.31374810703187447804778837341, −1.44162861834733743164888102913, 0.65466692722784065060143111047, 2.04039511351220061194192252184, 3.40735347305421600655551869686, 4.41455051044598582344335007730, 5.29054066127047546224443702035, 6.08350314838300969481819564883, 6.57959925477988174364963553365, 7.42378641780218111594438464394, 7.87887935859937297537962170146, 8.720060138017000965855156245705

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