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Properties

Label 2-1050-35.3-c1-0-15
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

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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: $\chi_{1050} (493, \cdot )$
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 \( 1 + (11.4 + 11.4i)T + 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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