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Properties

Label 2-1050-35.12-c1-0-16
Degree $2$
Conductor $1050$
Sign $0.101 + 0.994i$
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 + (1.86 + 1.87i)7-s + (0.707 − 0.707i)8-s + (−0.866 − 0.499i)9-s + (−2.74 − 4.75i)11-s + (−0.258 − 0.965i)12-s + (−2.41 − 2.41i)13-s + (2.28 + 1.33i)14-s + (0.500 − 0.866i)16-s + (2.04 + 0.548i)17-s + (−0.965 − 0.258i)18-s + (3.49 − 6.05i)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 + (0.704 + 0.709i)7-s + (0.249 − 0.249i)8-s + (−0.288 − 0.166i)9-s + (−0.827 − 1.43i)11-s + (−0.0747 − 0.278i)12-s + (−0.670 − 0.670i)13-s + (0.611 + 0.355i)14-s + (0.125 − 0.216i)16-s + (0.496 + 0.132i)17-s + (−0.227 − 0.0610i)18-s + (0.802 − 1.38i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.101 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.101 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.101 + 0.994i$
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: \((2,\ 1050,\ (\ :1/2),\ 0.101 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.551745892\)
\(L(\frac12)\) \(\approx\) \(2.551745892\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.86 - 1.87i)T \)
good11 \( 1 + (2.74 + 4.75i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.41 + 2.41i)T + 13iT^{2} \)
17 \( 1 + (-2.04 - 0.548i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3.49 + 6.05i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.454 - 1.69i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 0.684iT - 29T^{2} \)
31 \( 1 + (-4.82 + 2.78i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-9.46 + 2.53i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 2.50iT - 41T^{2} \)
43 \( 1 + (-1.95 + 1.95i)T - 43iT^{2} \)
47 \( 1 + (-0.912 - 3.40i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (9.08 + 2.43i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-5.08 - 8.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.01 - 0.585i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.57 - 9.61i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 11.9T + 71T^{2} \)
73 \( 1 + (1.26 - 4.70i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (7.21 + 4.16i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.05 + 4.05i)T + 83iT^{2} \)
89 \( 1 + (-3.59 + 6.22i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (13.1 - 13.1i)T - 97iT^{2} \)
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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

−9.757775467008027635909899553970, −8.745745585373469177380144596241, −7.939390950441739171462778025802, −7.32454414300901345554010246641, −5.97387767278532367651515126891, −5.52406165865229720068322216390, −4.60517073631331165695285406806, −3.01766685447928327091499255144, −2.59963457932178147730104921185, −0.945415671687156500710662852760, 1.76359946007866183602037838671, 2.96862810159535235425610653949, 4.24207334040928210232058593889, 4.70660702543065453591034625522, 5.55205157039750388337154719062, 6.81076132105072427568742356894, 7.65387450165319943900454229898, 8.105229943120893298273298887241, 9.598224589129384934973714285961, 10.05349580587155032622244065928

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