Properties

Label 2-3267-11.6-c0-0-2
Degree $2$
Conductor $3267$
Sign $0.322 - 0.946i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.809 + 0.587i)4-s + (1.13 + 1.56i)7-s + (0.492 − 0.159i)13-s + (0.309 − 0.951i)16-s + (0.831 − 1.14i)19-s + (0.809 + 0.587i)25-s + (−1.83 − 0.596i)28-s + (0.535 + 1.64i)31-s − 1.41i·43-s + (−0.844 + 2.59i)49-s + (−0.304 + 0.418i)52-s + (−1.34 − 0.437i)61-s + (0.309 + 0.951i)64-s − 1.73·67-s + (0.304 + 0.418i)73-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)4-s + (1.13 + 1.56i)7-s + (0.492 − 0.159i)13-s + (0.309 − 0.951i)16-s + (0.831 − 1.14i)19-s + (0.809 + 0.587i)25-s + (−1.83 − 0.596i)28-s + (0.535 + 1.64i)31-s − 1.41i·43-s + (−0.844 + 2.59i)49-s + (−0.304 + 0.418i)52-s + (−1.34 − 0.437i)61-s + (0.309 + 0.951i)64-s − 1.73·67-s + (0.304 + 0.418i)73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.322 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.322 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3267\)    =    \(3^{3} \cdot 11^{2}\)
Sign: $0.322 - 0.946i$
Analytic conductor: \(1.63044\)
Root analytic conductor: \(1.27688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3267} (2998, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3267,\ (\ :0),\ 0.322 - 0.946i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.236182304\)
\(L(\frac12)\) \(\approx\) \(1.236182304\)
\(L(1)\) 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 \( 1 + (-1.13 - 1.56i)T + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (-0.492 + 0.159i)T + (0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (-0.831 + 1.14i)T + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.535 - 1.64i)T + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 + 1.41iT - T^{2} \)
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 \( 1 + (1.34 + 0.437i)T + (0.809 + 0.587i)T^{2} \)
67 \( 1 + 1.73T + T^{2} \)
71 \( 1 + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.304 - 0.418i)T + (-0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.492 - 0.159i)T + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.809 + 0.587i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.989823672051613544856246502109, −8.391885694492574606477852346425, −7.69254626215024327712184032930, −6.83348811615590779726675734284, −5.66612911225605233875556011169, −5.09743961299624291393947322844, −4.58220999494675065451734370454, −3.31581641547828152460305261834, −2.65267263624233295962240111953, −1.36417890653897213981761366211, 0.924940453784855138517614883580, 1.66014619884992120719471127832, 3.32318865814808563111796291152, 4.38532399028027712552371183486, 4.49058098561877458183449576553, 5.61454609782385332099476287106, 6.32180314362712704775376814863, 7.39534717613930537514239017695, 7.952941532957029686974848945025, 8.557028932020514747988458097692

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