Properties

Label 2-300-300.287-c2-0-93
Degree $2$
Conductor $300$
Sign $0.798 + 0.602i$
Analytic cond. $8.17440$
Root an. cond. $2.85909$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.57 − 1.22i)2-s + (2.01 + 2.22i)3-s + (0.992 − 3.87i)4-s + (4.99 + 0.158i)5-s + (5.90 + 1.04i)6-s + (3.65 − 3.65i)7-s + (−3.18 − 7.33i)8-s + (−0.890 + 8.95i)9-s + (8.09 − 5.87i)10-s + (−9.99 − 7.26i)11-s + (10.6 − 5.59i)12-s + (2.08 − 13.1i)13-s + (1.29 − 10.2i)14-s + (9.71 + 11.4i)15-s + (−14.0 − 7.69i)16-s + (11.1 + 21.9i)17-s + ⋯
L(s)  = 1  + (0.789 − 0.613i)2-s + (0.671 + 0.741i)3-s + (0.248 − 0.968i)4-s + (0.999 + 0.0317i)5-s + (0.984 + 0.174i)6-s + (0.522 − 0.522i)7-s + (−0.397 − 0.917i)8-s + (−0.0989 + 0.995i)9-s + (0.809 − 0.587i)10-s + (−0.908 − 0.660i)11-s + (0.884 − 0.466i)12-s + (0.160 − 1.01i)13-s + (0.0924 − 0.733i)14-s + (0.647 + 0.762i)15-s + (−0.876 − 0.480i)16-s + (0.657 + 1.29i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.798 + 0.602i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.798 + 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.798 + 0.602i$
Analytic conductor: \(8.17440\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1),\ 0.798 + 0.602i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.37328 - 1.13017i\)
\(L(\frac12)\) \(\approx\) \(3.37328 - 1.13017i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.57 + 1.22i)T \)
3 \( 1 + (-2.01 - 2.22i)T \)
5 \( 1 + (-4.99 - 0.158i)T \)
good7 \( 1 + (-3.65 + 3.65i)T - 49iT^{2} \)
11 \( 1 + (9.99 + 7.26i)T + (37.3 + 115. i)T^{2} \)
13 \( 1 + (-2.08 + 13.1i)T + (-160. - 52.2i)T^{2} \)
17 \( 1 + (-11.1 - 21.9i)T + (-169. + 233. i)T^{2} \)
19 \( 1 + (9.91 - 30.5i)T + (-292. - 212. i)T^{2} \)
23 \( 1 + (-16.3 + 2.58i)T + (503. - 163. i)T^{2} \)
29 \( 1 + (4.19 + 12.9i)T + (-680. + 494. i)T^{2} \)
31 \( 1 + (28.6 + 9.30i)T + (777. + 564. i)T^{2} \)
37 \( 1 + (49.5 + 7.85i)T + (1.30e3 + 423. i)T^{2} \)
41 \( 1 + (-20.9 - 28.7i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 + (-27.3 - 27.3i)T + 1.84e3iT^{2} \)
47 \( 1 + (-15.8 + 31.1i)T + (-1.29e3 - 1.78e3i)T^{2} \)
53 \( 1 + (-20.6 + 40.5i)T + (-1.65e3 - 2.27e3i)T^{2} \)
59 \( 1 + (-0.366 - 0.504i)T + (-1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (61.1 + 44.4i)T + (1.14e3 + 3.53e3i)T^{2} \)
67 \( 1 + (-26.2 - 51.4i)T + (-2.63e3 + 3.63e3i)T^{2} \)
71 \( 1 + (-27.9 - 86.0i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (141. - 22.4i)T + (5.06e3 - 1.64e3i)T^{2} \)
79 \( 1 + (-2.59 - 7.97i)T + (-5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (36.7 + 72.2i)T + (-4.04e3 + 5.57e3i)T^{2} \)
89 \( 1 + (59.5 + 43.2i)T + (2.44e3 + 7.53e3i)T^{2} \)
97 \( 1 + (-26.0 + 51.0i)T + (-5.53e3 - 7.61e3i)T^{2} \)
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

−11.07390453392884429596548065569, −10.39024304064477042033220798759, −10.09477058252119637466682147789, −8.705920608242890362315539538855, −7.73565540523892372445766359280, −5.89614726665212085345909361286, −5.34837430751310057731076506015, −3.99658159784095986082522212136, −2.98385712707681091919569652134, −1.64906838633380219398099759474, 2.03166178897158721207584158245, 2.89822546076743180220503112432, 4.74938983219655147724117696647, 5.58822984939985264241154025323, 6.91375522615933326500381911813, 7.36522949071361922701735359834, 8.823741423063566845839906606828, 9.208869008823781727974015222621, 10.90332275639084145781371920699, 12.03848182841659516099619124403

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