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Properties

Label 2-1050-21.5-c1-0-34
Degree $2$
Conductor $1050$
Sign $-0.637 + 0.770i$
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  + (−0.866 − 0.5i)2-s + (−1.65 + 0.5i)3-s + (0.499 + 0.866i)4-s + (1.68 + 0.396i)6-s + (0.866 − 2.5i)7-s − 0.999i·8-s + (2.5 − 1.65i)9-s + (−0.813 + 0.469i)11-s + (−1.26 − 1.18i)12-s + 2i·13-s + (−2 + 1.73i)14-s + (−0.5 + 0.866i)16-s + (−3.31 − 5.74i)17-s + (−2.99 + 0.186i)18-s + (3 + 1.73i)19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.957 + 0.288i)3-s + (0.249 + 0.433i)4-s + (0.688 + 0.161i)6-s + (0.327 − 0.944i)7-s − 0.353i·8-s + (0.833 − 0.552i)9-s + (−0.245 + 0.141i)11-s + (−0.364 − 0.342i)12-s + 0.554i·13-s + (−0.534 + 0.462i)14-s + (−0.125 + 0.216i)16-s + (−0.804 − 1.39i)17-s + (−0.705 + 0.0438i)18-s + (0.688 + 0.397i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.637 + 0.770i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.637 + 0.770i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (551, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ -0.637 + 0.770i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5177559314\)
\(L(\frac12)\) \(\approx\) \(0.5177559314\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (1.65 - 0.5i)T \)
5 \( 1 \)
7 \( 1 + (-0.866 + 2.5i)T \)
good11 \( 1 + (0.813 - 0.469i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + (3.31 + 5.74i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3 - 1.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.18 + 0.686i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.31iT - 29T^{2} \)
31 \( 1 + (-6.55 + 3.78i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.10 - 7.11i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 7.37T + 41T^{2} \)
43 \( 1 - 1.08T + 43T^{2} \)
47 \( 1 + (-4.25 + 7.37i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.78 + 2.18i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.55 + 11.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (11.0 + 6.38i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.18 + 2.05i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.51iT - 71T^{2} \)
73 \( 1 + (-1.73 + i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.55 + 7.89i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 11.8T + 83T^{2} \)
89 \( 1 + (0.686 - 1.18i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 15.1iT - 97T^{2} \)
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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

−9.873612807314388902055178100966, −9.015456056602646199441645350811, −7.892434891226848336156743347720, −7.06385505004758330435722873482, −6.47582434181176964269548192840, −5.07710955098939272278833255138, −4.44989105036053424164311080351, −3.28832357694261597653119703954, −1.67391415860425800385335856940, −0.34734072287530648582804311464, 1.35678541842902681891765216691, 2.58644725695807302753928575422, 4.34113773717606327488230276537, 5.42605423031871921865119835776, 5.93352716735729068019312716298, 6.80060569994188316320123091166, 7.77301376510045671451602657295, 8.461976850669876649631033986074, 9.314689863536537228155520396861, 10.40462662630222552040887652821

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