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Properties

Label 2-3267-297.113-c0-0-0
Degree $2$
Conductor $3267$
Sign $-0.988 - 0.150i$
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  + (0.374 − 0.927i)3-s + (−0.882 − 0.469i)4-s + (−0.542 + 1.89i)5-s + (−0.719 − 0.694i)9-s + (−0.766 + 0.642i)12-s + (1.55 + 1.21i)15-s + (0.559 + 0.829i)16-s + (1.36 − 1.41i)20-s + (−1.70 − 0.300i)23-s + (−2.44 − 1.52i)25-s + (−0.913 + 0.406i)27-s + (−0.671 − 1.37i)31-s + (0.309 + 0.951i)36-s + (−0.232 − 0.258i)37-s + (1.70 − 0.984i)45-s + ⋯
L(s)  = 1  + (0.374 − 0.927i)3-s + (−0.882 − 0.469i)4-s + (−0.542 + 1.89i)5-s + (−0.719 − 0.694i)9-s + (−0.766 + 0.642i)12-s + (1.55 + 1.21i)15-s + (0.559 + 0.829i)16-s + (1.36 − 1.41i)20-s + (−1.70 − 0.300i)23-s + (−2.44 − 1.52i)25-s + (−0.913 + 0.406i)27-s + (−0.671 − 1.37i)31-s + (0.309 + 0.951i)36-s + (−0.232 − 0.258i)37-s + (1.70 − 0.984i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3267\)    =    \(3^{3} \cdot 11^{2}\)
Sign: $-0.988 - 0.150i$
Analytic conductor: \(1.63044\)
Root analytic conductor: \(1.27688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3267} (2786, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3267,\ (\ :0),\ -0.988 - 0.150i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04519643734\)
\(L(\frac12)\) \(\approx\) \(0.04519643734\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.374 + 0.927i)T \)
11 \( 1 \)
good2 \( 1 + (0.882 + 0.469i)T^{2} \)
5 \( 1 + (0.542 - 1.89i)T + (-0.848 - 0.529i)T^{2} \)
7 \( 1 + (0.961 + 0.275i)T^{2} \)
13 \( 1 + (0.990 + 0.139i)T^{2} \)
17 \( 1 + (-0.669 + 0.743i)T^{2} \)
19 \( 1 + (-0.104 - 0.994i)T^{2} \)
23 \( 1 + (1.70 + 0.300i)T + (0.939 + 0.342i)T^{2} \)
29 \( 1 + (0.719 - 0.694i)T^{2} \)
31 \( 1 + (0.671 + 1.37i)T + (-0.615 + 0.788i)T^{2} \)
37 \( 1 + (0.232 + 0.258i)T + (-0.104 + 0.994i)T^{2} \)
41 \( 1 + (0.719 + 0.694i)T^{2} \)
43 \( 1 + (0.766 + 0.642i)T^{2} \)
47 \( 1 + (-0.321 - 0.603i)T + (-0.559 + 0.829i)T^{2} \)
53 \( 1 + (-0.755 - 1.04i)T + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (1.28 - 0.0448i)T + (0.997 - 0.0697i)T^{2} \)
61 \( 1 + (-0.615 - 0.788i)T^{2} \)
67 \( 1 + (1.17 + 0.984i)T + (0.173 + 0.984i)T^{2} \)
71 \( 1 + (0.278 + 0.624i)T + (-0.669 + 0.743i)T^{2} \)
73 \( 1 + (0.913 - 0.406i)T^{2} \)
79 \( 1 + (-0.882 - 0.469i)T^{2} \)
83 \( 1 + (-0.990 + 0.139i)T^{2} \)
89 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (1.80 - 0.518i)T + (0.848 - 0.529i)T^{2} \)
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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

−8.120647554799933037995466130544, −7.77264892136879581961604283146, −7.00739933978817343240116885277, −6.15125789695300628384668773094, −5.83574502119993258235836196288, −4.27819190191630532555512585522, −3.64203059581268271444200054209, −2.74253671099325749754971100913, −1.84010405668305246494044459179, −0.02576460796519395051853795457, 1.62508953193746238429495315226, 3.22556132019212632215869645704, 4.03383807456461692156366025750, 4.43700038984622771974940066194, 5.21459587897563653451881810478, 5.70525730314851929544015929737, 7.37431797546830768770370943500, 8.213695814766061714417090871970, 8.450526665275352277249714900577, 9.103343717788894458027414638039

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