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Properties

Label 2-3267-297.106-c0-0-0
Degree $2$
Conductor $3267$
Sign $-0.0692 + 0.997i$
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  + (0.990 + 0.139i)3-s + (0.0348 − 0.999i)4-s + (0.454 − 1.82i)5-s + (0.961 + 0.275i)9-s + (0.173 − 0.984i)12-s + (0.704 − 1.74i)15-s + (−0.997 − 0.0697i)16-s + (−1.80 − 0.518i)20-s + (0.939 + 0.342i)23-s + (−2.23 − 1.18i)25-s + (0.913 + 0.406i)27-s + (0.194 + 0.287i)31-s + (0.309 − 0.951i)36-s + (−1.25 + 1.39i)37-s + (0.939 − 1.62i)45-s + ⋯
L(s)  = 1  + (0.990 + 0.139i)3-s + (0.0348 − 0.999i)4-s + (0.454 − 1.82i)5-s + (0.961 + 0.275i)9-s + (0.173 − 0.984i)12-s + (0.704 − 1.74i)15-s + (−0.997 − 0.0697i)16-s + (−1.80 − 0.518i)20-s + (0.939 + 0.342i)23-s + (−2.23 − 1.18i)25-s + (0.913 + 0.406i)27-s + (0.194 + 0.287i)31-s + (0.309 − 0.951i)36-s + (−1.25 + 1.39i)37-s + (0.939 − 1.62i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0692 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0692 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3267\)    =    \(3^{3} \cdot 11^{2}\)
Sign: $-0.0692 + 0.997i$
Analytic conductor: \(1.63044\)
Root analytic conductor: \(1.27688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3267} (403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3267,\ (\ :0),\ -0.0692 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.971977308\)
\(L(\frac12)\) \(\approx\) \(1.971977308\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.990 - 0.139i)T \)
11 \( 1 \)
good2 \( 1 + (-0.0348 + 0.999i)T^{2} \)
5 \( 1 + (-0.454 + 1.82i)T + (-0.882 - 0.469i)T^{2} \)
7 \( 1 + (0.241 - 0.970i)T^{2} \)
13 \( 1 + (0.615 + 0.788i)T^{2} \)
17 \( 1 + (-0.669 - 0.743i)T^{2} \)
19 \( 1 + (0.104 - 0.994i)T^{2} \)
23 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
29 \( 1 + (-0.961 + 0.275i)T^{2} \)
31 \( 1 + (-0.194 - 0.287i)T + (-0.374 + 0.927i)T^{2} \)
37 \( 1 + (1.25 - 1.39i)T + (-0.104 - 0.994i)T^{2} \)
41 \( 1 + (-0.961 - 0.275i)T^{2} \)
43 \( 1 + (-0.173 - 0.984i)T^{2} \)
47 \( 1 + (-0.0534 - 1.53i)T + (-0.997 + 0.0697i)T^{2} \)
53 \( 1 + (0.280 + 0.204i)T + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (-0.294 - 0.184i)T + (0.438 + 0.898i)T^{2} \)
61 \( 1 + (0.374 + 0.927i)T^{2} \)
67 \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \)
71 \( 1 + (-1.39 - 0.623i)T + (0.669 + 0.743i)T^{2} \)
73 \( 1 + (-0.913 - 0.406i)T^{2} \)
79 \( 1 + (-0.0348 + 0.999i)T^{2} \)
83 \( 1 + (0.615 - 0.788i)T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.370 + 1.48i)T + (-0.882 + 0.469i)T^{2} \)
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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

−8.678063200730594640412964938621, −8.254269113247129443673693896948, −7.21717864403634309094735058088, −6.28526761160269782514050306832, −5.34716969042629869594216899870, −4.84574645871670612233668669941, −4.18690823040190943129771773824, −2.89283988339430504202352128297, −1.71175294769462543877296490872, −1.15819903551258542318064340152, 2.07790270737296179375248539311, 2.57763971443856897560263582003, 3.47060251444047789592889008289, 3.81445656970418279318343639448, 5.18266890669974966807444216042, 6.50744607697839932761091587471, 6.87655427335687735965292533521, 7.48308863390744822080522128754, 8.156061109644642054599042736825, 8.983600044546104880292726361349

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