↑

Properties

Label 2-2808-9.4-c1-0-21
Degree $2$
Conductor $2808$
Sign $0.711 + 0.702i$
Analytic cond. $22.4219$
Root an. cond. $4.73518$
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  + (1.89 − 3.27i)5-s + (−0.659 − 1.14i)7-s + (2.20 + 3.81i)11-s + (−0.5 + 0.866i)13-s + 3.06·17-s + 7.13·19-s + (1.08 − 1.87i)23-s + (−4.64 − 8.04i)25-s + (4.17 + 7.22i)29-s + (−0.669 + 1.15i)31-s − 4.98·35-s + 1.24·37-s + (1.62 − 2.81i)41-s + (5.93 + 10.2i)43-s + (0.203 + 0.351i)47-s + ⋯
L(s)  = 1  + (0.845 − 1.46i)5-s + (−0.249 − 0.432i)7-s + (0.664 + 1.15i)11-s + (−0.138 + 0.240i)13-s + 0.742·17-s + 1.63·19-s + (0.226 − 0.391i)23-s + (−0.928 − 1.60i)25-s + (0.774 + 1.34i)29-s + (−0.120 + 0.208i)31-s − 0.843·35-s + 0.205·37-s + (0.253 − 0.439i)41-s + (0.904 + 1.56i)43-s + (0.0296 + 0.0513i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2808\)    =    \(2^{3} \cdot 3^{3} \cdot 13\)
Sign: $0.711 + 0.702i$
Analytic conductor: \(22.4219\)
Root analytic conductor: \(4.73518\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2808} (1873, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2808,\ (\ :1/2),\ 0.711 + 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.407685961\)
\(L(\frac12)\) \(\approx\) \(2.407685961\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-1.89 + 3.27i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.659 + 1.14i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.20 - 3.81i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 3.06T + 17T^{2} \)
19 \( 1 - 7.13T + 19T^{2} \)
23 \( 1 + (-1.08 + 1.87i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.17 - 7.22i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.669 - 1.15i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.24T + 37T^{2} \)
41 \( 1 + (-1.62 + 2.81i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.93 - 10.2i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.203 - 0.351i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 2.17T + 53T^{2} \)
59 \( 1 + (4.05 - 7.01i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.39 + 7.61i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.69 + 8.12i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.45T + 71T^{2} \)
73 \( 1 + 12.2T + 73T^{2} \)
79 \( 1 + (3.64 + 6.30i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.04 - 10.4i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 5.47T + 89T^{2} \)
97 \( 1 + (6.70 + 11.6i)T + (-48.5 + 84.0i)T^{2} \)
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

−8.949340738358611733172166232901, −7.956647800664982424338454484785, −7.20973465283503544166785630088, −6.40761507091114553403641269643, −5.43094339024917274958189937878, −4.88395378705346989861793372268, −4.14906025448989648251750586433, −2.96924751494402139449153453129, −1.62303129801248721527415485167, −1.01192831721700108739948641318, 1.09681274230208174380322394222, 2.51448736652701510146980998572, 3.07069465818127961707637499625, 3.84631453071304007085289020659, 5.40416888978949093620283180736, 5.86627373625615212104356431450, 6.48992246823765861587641856221, 7.32755589780489879784301135701, 8.003132661605016070527341360875, 9.142799752350380788969020151867

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