↑

Properties

Label 2-450-15.2-c3-0-10
Degree $2$
Conductor $450$
Sign $0.927 + 0.374i$
Analytic cond. $26.5508$
Root an. cond. $5.15275$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.41 + 1.41i)2-s − 4.00i·4-s + (9 + 9i)7-s + (5.65 + 5.65i)8-s − 38.1i·11-s + (−9 + 9i)13-s − 25.4·14-s − 16.0·16-s + (55.1 − 55.1i)17-s + 38i·19-s + (54.0 + 54.0i)22-s + (−29.6 − 29.6i)23-s − 25.4i·26-s + (36.0 − 36.0i)28-s − 76.3·29-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (0.485 + 0.485i)7-s + (0.250 + 0.250i)8-s − 1.04i·11-s + (−0.192 + 0.192i)13-s − 0.485·14-s − 0.250·16-s + (0.786 − 0.786i)17-s + 0.458i·19-s + (0.523 + 0.523i)22-s + (−0.269 − 0.269i)23-s − 0.192i·26-s + (0.242 − 0.242i)28-s − 0.489·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.927 + 0.374i$
Analytic conductor: \(26.5508\)
Root analytic conductor: \(5.15275\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :3/2),\ 0.927 + 0.374i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.379846100\)
\(L(\frac12)\) \(\approx\) \(1.379846100\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 - 1.41i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-9 - 9i)T + 343iT^{2} \)
11 \( 1 + 38.1iT - 1.33e3T^{2} \)
13 \( 1 + (9 - 9i)T - 2.19e3iT^{2} \)
17 \( 1 + (-55.1 + 55.1i)T - 4.91e3iT^{2} \)
19 \( 1 - 38iT - 6.85e3T^{2} \)
23 \( 1 + (29.6 + 29.6i)T + 1.21e4iT^{2} \)
29 \( 1 + 76.3T + 2.43e4T^{2} \)
31 \( 1 + 236T + 2.97e4T^{2} \)
37 \( 1 + (-279 - 279i)T + 5.06e4iT^{2} \)
41 \( 1 + 343. iT - 6.89e4T^{2} \)
43 \( 1 + (-360 + 360i)T - 7.95e4iT^{2} \)
47 \( 1 + (-254. + 254. i)T - 1.03e5iT^{2} \)
53 \( 1 + (352. + 352. i)T + 1.48e5iT^{2} \)
59 \( 1 - 343.T + 2.05e5T^{2} \)
61 \( 1 - 110T + 2.26e5T^{2} \)
67 \( 1 + (-702 - 702i)T + 3.00e5iT^{2} \)
71 \( 1 + 76.3iT - 3.57e5T^{2} \)
73 \( 1 + (-378 + 378i)T - 3.89e5iT^{2} \)
79 \( 1 + 272iT - 4.93e5T^{2} \)
83 \( 1 + (-59.3 - 59.3i)T + 5.71e5iT^{2} \)
89 \( 1 - 1.48e3T + 7.04e5T^{2} \)
97 \( 1 + (-684 - 684i)T + 9.12e5iT^{2} \)
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

−10.55462503593424941551767064463, −9.545429607455697481800076695778, −8.739157383317319891024668737792, −7.954484813970587404542835961336, −7.04882283229206969776501894629, −5.83232542522215860696628924083, −5.21243696309118581781305025293, −3.68420149007829570962423611275, −2.17226842668786829232489012171, −0.62315063673353438540020686529, 1.11270298129949158169903295171, 2.31529501137156449941105429341, 3.77020217718784912211662796744, 4.73983843770761698555538361825, 6.06995786070420278357598525718, 7.52373376329140275817398132052, 7.77048723665687636191416739286, 9.181876600883300510866638706086, 9.794192873698598457535222835043, 10.79567903452824322565423050329

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