Properties

Label 2-4032-8.5-c1-0-22
Degree $2$
Conductor $4032$
Sign $0.707 - 0.707i$
Analytic cond. $32.1956$
Root an. cond. $5.67412$
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  + 2i·5-s + 7-s − 4i·11-s + 2i·13-s + 2·17-s − 6·23-s + 25-s + 2i·29-s + 4·31-s + 2i·35-s + 4i·37-s + 10·41-s + 2i·43-s + 8·47-s + 49-s + ⋯
L(s)  = 1  + 0.894i·5-s + 0.377·7-s − 1.20i·11-s + 0.554i·13-s + 0.485·17-s − 1.25·23-s + 0.200·25-s + 0.371i·29-s + 0.718·31-s + 0.338i·35-s + 0.657i·37-s + 1.56·41-s + 0.304i·43-s + 1.16·47-s + 0.142·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(32.1956\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4032} (2017, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4032,\ (\ :1/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.958212049\)
\(L(\frac12)\) \(\approx\) \(1.958212049\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 - T \)
good5 \( 1 - 2iT - 5T^{2} \)
11 \( 1 + 4iT - 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 - 2iT - 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 4iT - 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 - 2iT - 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 4iT - 59T^{2} \)
61 \( 1 - 10iT - 61T^{2} \)
67 \( 1 - 2iT - 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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

−8.471838962770586415948708454519, −7.80381142093737616370992922728, −7.08969589350899637640021816754, −6.23009748953773225726142378952, −5.80251375175720343170288047653, −4.73007558741843598847849267318, −3.84443787968040537033968804552, −3.06740082358320647091756732469, −2.22795057727428517874093116653, −0.950956034959552966560564407694, 0.70999647413327453779454816643, 1.78423328056731611659488669630, 2.69463069805405947864151000994, 4.02328660278433705918852372644, 4.50793801986987985479109833042, 5.35845568595979807839408016799, 5.96576646339378720692679316871, 7.01686402147851736568768611083, 7.78688872305237803604980339164, 8.200686335427796865503521379987

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