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Properties

Label 2-3267-33.26-c0-0-2
Degree $2$
Conductor $3267$
Sign $-0.138 - 0.990i$
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  + (0.309 + 0.951i)4-s + (0.535 + 1.64i)7-s + (1.40 + 1.01i)13-s + (−0.809 + 0.587i)16-s + (0.309 − 0.951i)25-s + (−1.40 + 1.01i)28-s + (−0.809 − 0.587i)31-s + (−0.618 − 1.90i)37-s + (−1.61 + 1.17i)49-s + (−0.535 + 1.64i)52-s + (−0.809 − 0.587i)64-s + 67-s + (−0.535 − 1.64i)73-s + (1.40 + 1.01i)79-s + (−0.927 + 2.85i)91-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)4-s + (0.535 + 1.64i)7-s + (1.40 + 1.01i)13-s + (−0.809 + 0.587i)16-s + (0.309 − 0.951i)25-s + (−1.40 + 1.01i)28-s + (−0.809 − 0.587i)31-s + (−0.618 − 1.90i)37-s + (−1.61 + 1.17i)49-s + (−0.535 + 1.64i)52-s + (−0.809 − 0.587i)64-s + 67-s + (−0.535 − 1.64i)73-s + (1.40 + 1.01i)79-s + (−0.927 + 2.85i)91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.138 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.138 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3267\)    =    \(3^{3} \cdot 11^{2}\)
Sign: $-0.138 - 0.990i$
Analytic conductor: \(1.63044\)
Root analytic conductor: \(1.27688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3267} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3267,\ (\ :0),\ -0.138 - 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.490400073\)
\(L(\frac12)\) \(\approx\) \(1.490400073\)
\(L(1)\) 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 \( 1 + (-0.309 - 0.951i)T^{2} \)
5 \( 1 + (-0.309 + 0.951i)T^{2} \)
7 \( 1 + (-0.535 - 1.64i)T + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (-1.40 - 1.01i)T + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (-0.809 - 0.587i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.618 + 1.90i)T + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.309 - 0.951i)T^{2} \)
67 \( 1 - T + T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.535 + 1.64i)T + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (-1.40 - 1.01i)T + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.309 + 0.951i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
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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

−8.945311217834349208868818473191, −8.374241769567576491051360002511, −7.71084183238227635668850863784, −6.71035219437790250767442759135, −6.09548617614323895831711200701, −5.30535490156922867749979685520, −4.25514532843930860726653968210, −3.52883593992294850045538045890, −2.43162509362731823549153244251, −1.84334850771038186060414741158, 0.977797967273941550486679143374, 1.58432429783273066992307798577, 3.15062700987905488767121907155, 3.90942786077673809977453538883, 4.90814441648953419554928809696, 5.51541139371785603852964447789, 6.49138156645023026973377015922, 7.02774931959390656341018512922, 7.84817114424950090257157553570, 8.531219560914317117845223954068

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