↑

Properties

Label 2-2904-264.107-c0-0-7
Degree $2$
Conductor $2904$
Sign $-0.822 + 0.568i$
Analytic cond. $1.44928$
Root an. cond. $1.20386$
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)2-s − 3-s + (0.309 − 0.951i)4-s + (0.809 − 0.587i)6-s + (0.309 + 0.951i)8-s + 9-s + (−0.309 + 0.951i)12-s + (−0.809 − 0.587i)16-s + (−1.30 − 0.951i)17-s + (−0.809 + 0.587i)18-s + (−1.11 + 0.363i)19-s + (−0.309 − 0.951i)24-s + (−0.309 − 0.951i)25-s − 27-s + 32-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)2-s − 3-s + (0.309 − 0.951i)4-s + (0.809 − 0.587i)6-s + (0.309 + 0.951i)8-s + 9-s + (−0.309 + 0.951i)12-s + (−0.809 − 0.587i)16-s + (−1.30 − 0.951i)17-s + (−0.809 + 0.587i)18-s + (−1.11 + 0.363i)19-s + (−0.309 − 0.951i)24-s + (−0.309 − 0.951i)25-s − 27-s + 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.822 + 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.822 + 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2904\)    =    \(2^{3} \cdot 3 \cdot 11^{2}\)
Sign: $-0.822 + 0.568i$
Analytic conductor: \(1.44928\)
Root analytic conductor: \(1.20386\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2904} (1691, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2904,\ (\ :0),\ -0.822 + 0.568i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04590985709\)
\(L(\frac12)\) \(\approx\) \(0.04590985709\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 - 0.587i)T \)
3 \( 1 + T \)
11 \( 1 \)
good5 \( 1 + (0.309 + 0.951i)T^{2} \)
7 \( 1 + (-0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.309 - 0.951i)T^{2} \)
17 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
19 \( 1 + (1.11 - 0.363i)T + (0.809 - 0.587i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.809 + 0.587i)T^{2} \)
41 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
43 \( 1 - 1.90iT - T^{2} \)
47 \( 1 + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (1.11 + 0.363i)T + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.309 + 0.951i)T^{2} \)
67 \( 1 + 1.61T + T^{2} \)
71 \( 1 + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (1.80 + 0.587i)T + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (0.309 - 0.951i)T^{2} \)
83 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
89 \( 1 + 1.90iT - T^{2} \)
97 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
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

−8.686276392058015541460462634107, −7.76373721343400523669376329761, −7.12644891968222602267213164903, −6.21584011591731406317557359279, −6.04942519408465391745292985242, −4.73455257230042286814772817021, −4.42575975113177876385229634555, −2.61575799208321475271586199091, −1.50483111317085854766105016642, −0.04316559917529121968754823266, 1.52600127436587035706300352040, 2.39435986503067165148850543661, 3.83249877999268362330417788055, 4.34984378985614094639882490440, 5.49425502270461978814483586335, 6.41413732747031582417091853310, 6.98500474520763319189838231212, 7.74957108422498249386615768012, 8.791118779464791855979044868827, 9.123460120365603689427116388137

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