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Properties

Label 2-3267-297.112-c0-0-0
Degree $2$
Conductor $3267$
Sign $-0.375 + 0.926i$
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  + (0.559 − 0.829i)3-s + (−0.241 − 0.970i)4-s + (1.51 − 0.213i)5-s + (−0.374 − 0.927i)9-s + (−0.939 − 0.342i)12-s + (0.671 − 1.37i)15-s + (−0.882 + 0.469i)16-s + (−0.573 − 1.42i)20-s + (−0.766 − 0.642i)23-s + (1.29 − 0.371i)25-s + (−0.978 − 0.207i)27-s + (−1.59 + 0.995i)31-s + (−0.809 + 0.587i)36-s + (1.39 − 0.623i)37-s + (−0.766 − 1.32i)45-s + ⋯
L(s)  = 1  + (0.559 − 0.829i)3-s + (−0.241 − 0.970i)4-s + (1.51 − 0.213i)5-s + (−0.374 − 0.927i)9-s + (−0.939 − 0.342i)12-s + (0.671 − 1.37i)15-s + (−0.882 + 0.469i)16-s + (−0.573 − 1.42i)20-s + (−0.766 − 0.642i)23-s + (1.29 − 0.371i)25-s + (−0.978 − 0.207i)27-s + (−1.59 + 0.995i)31-s + (−0.809 + 0.587i)36-s + (1.39 − 0.623i)37-s + (−0.766 − 1.32i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.375 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.375 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3267\)    =    \(3^{3} \cdot 11^{2}\)
Sign: $-0.375 + 0.926i$
Analytic conductor: \(1.63044\)
Root analytic conductor: \(1.27688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3267} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3267,\ (\ :0),\ -0.375 + 0.926i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.786408487\)
\(L(\frac12)\) \(\approx\) \(1.786408487\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.559 + 0.829i)T \)
11 \( 1 \)
good2 \( 1 + (0.241 + 0.970i)T^{2} \)
5 \( 1 + (-1.51 + 0.213i)T + (0.961 - 0.275i)T^{2} \)
7 \( 1 + (-0.990 + 0.139i)T^{2} \)
13 \( 1 + (0.997 - 0.0697i)T^{2} \)
17 \( 1 + (-0.913 - 0.406i)T^{2} \)
19 \( 1 + (-0.669 - 0.743i)T^{2} \)
23 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
29 \( 1 + (0.374 - 0.927i)T^{2} \)
31 \( 1 + (1.59 - 0.995i)T + (0.438 - 0.898i)T^{2} \)
37 \( 1 + (-1.39 + 0.623i)T + (0.669 - 0.743i)T^{2} \)
41 \( 1 + (0.374 + 0.927i)T^{2} \)
43 \( 1 + (0.939 - 0.342i)T^{2} \)
47 \( 1 + (0.0840 - 0.336i)T + (-0.882 - 0.469i)T^{2} \)
53 \( 1 + (0.580 - 1.78i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (-1.35 + 1.30i)T + (0.0348 - 0.999i)T^{2} \)
61 \( 1 + (-0.438 - 0.898i)T^{2} \)
67 \( 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2} \)
71 \( 1 + (0.339 + 0.0722i)T + (0.913 + 0.406i)T^{2} \)
73 \( 1 + (0.978 + 0.207i)T^{2} \)
79 \( 1 + (0.241 + 0.970i)T^{2} \)
83 \( 1 + (0.997 + 0.0697i)T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.343 - 0.0483i)T + (0.961 + 0.275i)T^{2} \)
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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

−8.835754149481946115670612738028, −7.899523226743529627661256063010, −6.91560843273602982136007151691, −6.25488260425500558222685199277, −5.74085828717155661688812799077, −5.02861753511235055719268600091, −3.86734588228430920674782758103, −2.49905087658035311484361147937, −1.94121503781780620430441310535, −1.02598634844592131122544471573, 1.98202009993201904432425604784, 2.60339973539211482239120303861, 3.57038134160755737689215600815, 4.25799779548341582951235670198, 5.29133246429576600652105896004, 5.84548244519616445426225303525, 6.90400759651304721491827617923, 7.73021452716980329804538863970, 8.445045002589601251475997223408, 9.164845385702501856446375828669

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