↑

Properties

Label 2-3312-207.160-c0-0-2
Degree $2$
Conductor $3312$
Sign $-0.5 + 0.866i$
Analytic cond. $1.65290$
Root an. cond. $1.28565$
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  + (−0.766 − 0.642i)3-s + (0.173 + 0.984i)9-s + (−0.766 − 1.32i)13-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s + (0.500 − 0.866i)27-s + (0.939 − 1.62i)29-s + (−0.939 − 1.62i)31-s + (−0.266 + 1.50i)39-s + (−0.173 − 0.300i)41-s + (0.173 − 0.300i)47-s + (−0.5 − 0.866i)49-s + (−0.5 − 0.866i)59-s + (0.173 − 0.984i)69-s + 1.87·71-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)3-s + (0.173 + 0.984i)9-s + (−0.766 − 1.32i)13-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s + (0.500 − 0.866i)27-s + (0.939 − 1.62i)29-s + (−0.939 − 1.62i)31-s + (−0.266 + 1.50i)39-s + (−0.173 − 0.300i)41-s + (0.173 − 0.300i)47-s + (−0.5 − 0.866i)49-s + (−0.5 − 0.866i)59-s + (0.173 − 0.984i)69-s + 1.87·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3312\)    =    \(2^{4} \cdot 3^{2} \cdot 23\)
Sign: $-0.5 + 0.866i$
Analytic conductor: \(1.65290\)
Root analytic conductor: \(1.28565\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3312} (3265, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3312,\ (\ :0),\ -0.5 + 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6842030933\)
\(L(\frac12)\) \(\approx\) \(0.6842030933\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.766 + 0.642i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
29 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.939 + 1.62i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - 1.87T + T^{2} \)
73 \( 1 + 1.87T + T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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

−8.269990978202875306561264204210, −7.70531723446236818058969416207, −7.19647530115848783465124195148, −6.21777205977920707838943791009, −5.56051320067499042150824832532, −5.00847003664686549122591045346, −3.91708101081398597790005577963, −2.79123495075364308532954693092, −1.83570701715485786169430284921, −0.46698019097156645973380625799, 1.38091654251952449792719728941, 2.71328754233653665205258922930, 3.73244920145056174546961957911, 4.70169740180242502520211415921, 4.97620770230451288103735839350, 6.12747999151061047139605696323, 6.73983787942239903431629116054, 7.32815135899593044910188833904, 8.605086363794085639190672856554, 9.059300683453926526952554073686

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