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Properties

Label 2-3267-33.5-c0-0-4
Degree $2$
Conductor $3267$
Sign $0.955 - 0.296i$
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  + (−0.809 + 0.587i)4-s + (1.40 − 1.01i)7-s + (0.535 + 1.64i)13-s + (0.309 − 0.951i)16-s + (−0.809 − 0.587i)25-s + (−0.535 + 1.64i)28-s + (0.309 + 0.951i)31-s + (1.61 − 1.17i)37-s + (0.618 − 1.90i)49-s + (−1.40 − 1.01i)52-s + (0.309 + 0.951i)64-s + 67-s + (−1.40 + 1.01i)73-s + (0.535 + 1.64i)79-s + (2.42 + 1.76i)91-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)4-s + (1.40 − 1.01i)7-s + (0.535 + 1.64i)13-s + (0.309 − 0.951i)16-s + (−0.809 − 0.587i)25-s + (−0.535 + 1.64i)28-s + (0.309 + 0.951i)31-s + (1.61 − 1.17i)37-s + (0.618 − 1.90i)49-s + (−1.40 − 1.01i)52-s + (0.309 + 0.951i)64-s + 67-s + (−1.40 + 1.01i)73-s + (0.535 + 1.64i)79-s + (2.42 + 1.76i)91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3267\)    =    \(3^{3} \cdot 11^{2}\)
Sign: $0.955 - 0.296i$
Analytic conductor: \(1.63044\)
Root analytic conductor: \(1.27688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3267} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3267,\ (\ :0),\ 0.955 - 0.296i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.255285679\)
\(L(\frac12)\) \(\approx\) \(1.255285679\)
\(L(1)\) 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 \( 1 + (0.809 - 0.587i)T^{2} \)
5 \( 1 + (0.809 + 0.587i)T^{2} \)
7 \( 1 + (-1.40 + 1.01i)T + (0.309 - 0.951i)T^{2} \)
13 \( 1 + (-0.535 - 1.64i)T + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (0.309 + 0.951i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (-1.61 + 1.17i)T + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.809 - 0.587i)T^{2} \)
67 \( 1 - T + T^{2} \)
71 \( 1 + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (1.40 - 1.01i)T + (0.309 - 0.951i)T^{2} \)
79 \( 1 + (-0.535 - 1.64i)T + (-0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.809 + 0.587i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
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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

−8.700957476592016026658986608445, −8.187837929054881090594858745267, −7.47868446948580842570471937541, −6.84283839566174265653339136566, −5.74362423057227124862179632987, −4.67185386896563118855272927297, −4.30582770921797044786264947756, −3.66774085737244551595912799847, −2.22009894098205328378794168254, −1.12552068004285679721556097018, 1.03295609130040096468983528000, 2.09238739695922480170274799493, 3.22801539631462225947397688744, 4.36781376514039300401291353865, 5.02778685456962404870483034177, 5.72855852958456298897652638646, 6.12677664425773923615190612087, 7.72757857923463652781193486595, 8.064938564892446364358928312464, 8.715446002310813345736983270521

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