↑

Properties

Label 2-1350-45.23-c1-0-9
Degree $2$
Conductor $1350$
Sign $0.860 + 0.509i$
Analytic cond. $10.7798$
Root an. cond. $3.28326$
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  + (−0.258 + 0.965i)2-s + (−0.866 − 0.499i)4-s + (−2.32 − 0.622i)7-s + (0.707 − 0.707i)8-s + (−0.991 + 0.572i)11-s + (−2.38 + 0.640i)13-s + (1.20 − 2.08i)14-s + (0.500 + 0.866i)16-s + (4.99 + 4.99i)17-s − 2.78i·19-s + (−0.296 − 1.10i)22-s + (−1.59 − 5.95i)23-s − 2.47i·26-s + (1.70 + 1.70i)28-s + (0.672 + 1.16i)29-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.433 − 0.249i)4-s + (−0.877 − 0.235i)7-s + (0.249 − 0.249i)8-s + (−0.299 + 0.172i)11-s + (−0.662 + 0.177i)13-s + (0.321 − 0.556i)14-s + (0.125 + 0.216i)16-s + (1.21 + 1.21i)17-s − 0.638i·19-s + (−0.0631 − 0.235i)22-s + (−0.332 − 1.24i)23-s − 0.485i·26-s + (0.321 + 0.321i)28-s + (0.124 + 0.216i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.509i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.860 + 0.509i$
Analytic conductor: \(10.7798\)
Root analytic conductor: \(3.28326\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1350} (1043, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ 0.860 + 0.509i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9459674438\)
\(L(\frac12)\) \(\approx\) \(0.9459674438\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (2.32 + 0.622i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (0.991 - 0.572i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.38 - 0.640i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-4.99 - 4.99i)T + 17iT^{2} \)
19 \( 1 + 2.78iT - 19T^{2} \)
23 \( 1 + (1.59 + 5.95i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-0.672 - 1.16i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.25 + 2.16i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-8.16 + 8.16i)T - 37iT^{2} \)
41 \( 1 + (-1.70 - 0.986i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.32 + 8.68i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-3.19 + 11.9i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.84 - 1.84i)T - 53iT^{2} \)
59 \( 1 + (-1.31 + 2.27i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.54 + 6.13i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.0146 - 0.0545i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 9.10iT - 71T^{2} \)
73 \( 1 + (7.82 + 7.82i)T + 73iT^{2} \)
79 \( 1 + (-8.46 + 4.88i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.70 - 0.724i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + 4.87T + 89T^{2} \)
97 \( 1 + (-7.79 - 2.08i)T + (84.0 + 48.5i)T^{2} \)
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

−9.533468267629635103696634587636, −8.672442945699716128028137844949, −7.82519513500026981190937770052, −7.10813505235774209302195662230, −6.29435128650414812665841789704, −5.56166611730344456126331541076, −4.48965580996614161471846367626, −3.56708114183726810105204356646, −2.28797260901112600099034218567, −0.47822492912447458098952994336, 1.13212289046043470549965188524, 2.73576362122136440678106740172, 3.22004124741842443206339374284, 4.47776180380717638367472034957, 5.46593999309944969172829061271, 6.27710220793516748500553527978, 7.54851864047555482366837985062, 7.950276086072170482530541634601, 9.267107839731366454252046040096, 9.688251534299028017907538190193

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