Properties

Label 2-3267-99.76-c0-0-1
Degree $2$
Conductor $3267$
Sign $0.908 + 0.418i$
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  + (1.22 − 0.707i)2-s + (0.499 − 0.866i)4-s + (−0.5 + 0.866i)5-s + (1.22 − 0.707i)7-s + 1.41i·10-s + (0.999 − 1.73i)14-s + (0.499 + 0.866i)16-s + (0.5 + 0.866i)20-s − 1.41i·28-s + (1.22 − 0.707i)29-s + (−0.5 + 0.866i)31-s + (1.22 + 0.707i)32-s + 1.41i·35-s − 37-s + (−0.5 − 0.866i)47-s + ⋯
L(s)  = 1  + (1.22 − 0.707i)2-s + (0.499 − 0.866i)4-s + (−0.5 + 0.866i)5-s + (1.22 − 0.707i)7-s + 1.41i·10-s + (0.999 − 1.73i)14-s + (0.499 + 0.866i)16-s + (0.5 + 0.866i)20-s − 1.41i·28-s + (1.22 − 0.707i)29-s + (−0.5 + 0.866i)31-s + (1.22 + 0.707i)32-s + 1.41i·35-s − 37-s + (−0.5 − 0.866i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3267\)    =    \(3^{3} \cdot 11^{2}\)
Sign: $0.908 + 0.418i$
Analytic conductor: \(1.63044\)
Root analytic conductor: \(1.27688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3267} (604, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3267,\ (\ :0),\ 0.908 + 0.418i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.542713085\)
\(L(\frac12)\) \(\approx\) \(2.542713085\)
\(L(1)\) 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 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 - 1.41iT - T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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

−8.478499749345783567591102676052, −8.073498837148660099530167640303, −7.12348144832413339363008263061, −6.53063123151449472267874977871, −5.32525680514042746651559890159, −4.84096493803381442125342042960, −3.96882884523849774906428846481, −3.41200343143198147628981986891, −2.45713765439714152096810372335, −1.46751715252266370944596190580, 1.27036646009138617738167413137, 2.58477148578749871926585032246, 3.74123514913048708029816218920, 4.53430283961575243409377519814, 5.01935891409478439701681690436, 5.58021581971990007543261638050, 6.45949723561087218192974298514, 7.31580076194902298022313467751, 8.102636883244244227897888418841, 8.557317386978878943800450016542

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