↑

Properties

Label 2-1050-35.12-c1-0-0
Degree $2$
Conductor $1050$
Sign $-0.226 - 0.974i$
Analytic cond. $8.38429$
Root an. cond. $2.89556$
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  + (−0.965 + 0.258i)2-s + (0.258 − 0.965i)3-s + (0.866 − 0.499i)4-s + i·6-s + (−2.63 − 0.258i)7-s + (−0.707 + 0.707i)8-s + (−0.866 − 0.499i)9-s + (1.17 + 2.04i)11-s + (−0.258 − 0.965i)12-s + (0.0968 + 0.0968i)13-s + (2.61 − 0.431i)14-s + (0.500 − 0.866i)16-s + (−3.11 − 0.833i)17-s + (0.965 + 0.258i)18-s + (−0.434 + 0.752i)19-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.149 − 0.557i)3-s + (0.433 − 0.249i)4-s + 0.408i·6-s + (−0.995 − 0.0978i)7-s + (−0.249 + 0.249i)8-s + (−0.288 − 0.166i)9-s + (0.355 + 0.615i)11-s + (−0.0747 − 0.278i)12-s + (0.0268 + 0.0268i)13-s + (0.697 − 0.115i)14-s + (0.125 − 0.216i)16-s + (−0.754 − 0.202i)17-s + (0.227 + 0.0610i)18-s + (−0.0996 + 0.172i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.226 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.226 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.226 - 0.974i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ -0.226 - 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5259251256\)
\(L(\frac12)\) \(\approx\) \(0.5259251256\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 \)
7 \( 1 + (2.63 + 0.258i)T \)
good11 \( 1 + (-1.17 - 2.04i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.0968 - 0.0968i)T + 13iT^{2} \)
17 \( 1 + (3.11 + 0.833i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.434 - 0.752i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.833 - 3.11i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 4.08iT - 29T^{2} \)
31 \( 1 + (0.747 - 0.431i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (9.51 - 2.54i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 6.86iT - 41T^{2} \)
43 \( 1 + (2.57 - 2.57i)T - 43iT^{2} \)
47 \( 1 + (0.223 + 0.833i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-8.07 - 2.16i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-5.35 - 9.27i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.44 + 4.29i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.28 - 12.2i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 5.13T + 71T^{2} \)
73 \( 1 + (-4.20 + 15.7i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (4.02 + 2.32i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.13 - 7.13i)T + 83iT^{2} \)
89 \( 1 + (4.98 - 8.63i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (11.4 - 11.4i)T - 97iT^{2} \)
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

−9.990784976984925511900450173601, −9.226115827716238522499713111872, −8.629708988974667321702235891565, −7.53732090363414458536431408132, −6.87627978036826328989910858207, −6.30561911435558494845958858692, −5.14794624542592895915996158790, −3.75442477652222610550032249720, −2.64449560318579642161326065986, −1.40106872559836962494048353733, 0.29072745843822968042676068886, 2.20121406318917581670815534721, 3.29231387806716864476161639420, 4.11044668287393485113039079311, 5.48021830838406783247550735611, 6.43127120259519274530064148415, 7.13574480449002748680491719331, 8.423019105711328317946115657627, 8.878227982421156917847786777920, 9.642112908909198822454568576616

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