Properties

Label 2-4056-13.12-c1-0-0
Degree $2$
Conductor $4056$
Sign $-0.554 + 0.832i$
Analytic cond. $32.3873$
Root an. cond. $5.69098$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 4i·5-s + 9-s + 2i·11-s − 4i·15-s − 2·17-s + 8i·19-s − 4·23-s − 11·25-s − 27-s − 6·29-s − 4i·31-s − 2i·33-s − 6i·37-s − 12i·41-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.78i·5-s + 0.333·9-s + 0.603i·11-s − 1.03i·15-s − 0.485·17-s + 1.83i·19-s − 0.834·23-s − 2.20·25-s − 0.192·27-s − 1.11·29-s − 0.718i·31-s − 0.348i·33-s − 0.986i·37-s − 1.87i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4056\)    =    \(2^{3} \cdot 3 \cdot 13^{2}\)
Sign: $-0.554 + 0.832i$
Analytic conductor: \(32.3873\)
Root analytic conductor: \(5.69098\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4056} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4056,\ (\ :1/2),\ -0.554 + 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3722574923\)
\(L(\frac12)\) \(\approx\) \(0.3722574923\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
13 \( 1 \)
good5 \( 1 - 4iT - 5T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 8iT - 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 4iT - 31T^{2} \)
37 \( 1 + 6iT - 37T^{2} \)
41 \( 1 + 12iT - 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 - 6iT - 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 - 14iT - 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 - 2iT - 71T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 14iT - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 10iT - 97T^{2} \)
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

−8.997288635621098660284911847112, −7.82437074055018300681173828612, −7.39494316028735243940451077596, −6.74724172232196270335050684510, −5.93812720205782917887906178064, −5.56173143830816768412169856756, −4.04026961310307234749478073424, −3.77910002239430858907043900061, −2.49689490673139884913913650682, −1.82773280301130631086205581584, 0.12714425726434420776516908787, 1.02532474988957540364259615939, 2.05941343744698439545371308178, 3.42141294039053688788336290104, 4.48536748280519740277606260594, 4.88930915865677718139200468288, 5.58206465061934951192235524288, 6.37311257314733053783950046564, 7.18914116641740289640708927112, 8.226660941077146572274547627181

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