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Properties

Label 2-1050-21.5-c1-0-1
Degree $2$
Conductor $1050$
Sign $-0.957 + 0.289i$
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 + (−1.18 − 1.26i)6-s + (−0.866 + 2.5i)7-s + 0.999i·8-s + (2.5 + 1.65i)9-s + (−3.68 + 2.12i)11-s + (−0.396 − 1.68i)12-s − 2i·13-s + (−2 + 1.73i)14-s + (−0.5 + 0.866i)16-s + (−3.31 − 5.74i)17-s + (1.33 + 2.68i)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 + (−0.484 − 0.515i)6-s + (−0.327 + 0.944i)7-s + 0.353i·8-s + (0.833 + 0.552i)9-s + (−1.11 + 0.641i)11-s + (−0.114 − 0.486i)12-s − 0.554i·13-s + (−0.534 + 0.462i)14-s + (−0.125 + 0.216i)16-s + (−0.804 − 1.39i)17-s + (0.314 + 0.633i)18-s + (0.688 + 0.397i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.957 + 0.289i$
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: \((2,\ 1050,\ (\ :1/2),\ -0.957 + 0.289i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2840391482\)
\(L(\frac12)\) \(\approx\) \(0.2840391482\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.68 - 2.12i)T + (5.5 - 9.52i)T^{2} \)
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 \( 1 + (3.78 + 2.18i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.31iT - 29T^{2} \)
31 \( 1 + (2.05 - 1.18i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.84 - 10.1i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.62T + 41T^{2} \)
43 \( 1 + 11.0T + 43T^{2} \)
47 \( 1 + (0.939 - 1.62i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.18 + 0.686i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.05 - 3.56i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.44 + 1.40i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.78 + 6.55i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.87iT - 71T^{2} \)
73 \( 1 + (1.73 - i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.05 - 7.02i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.43T + 83T^{2} \)
89 \( 1 + (-2.18 + 3.78i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2.11iT - 97T^{2} \)
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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

−10.33088512537031903307467689851, −9.777915201170384903000671229465, −8.492243743081096560133009591164, −7.63913640385959028620243048551, −6.84649248250504357091445534813, −6.02994580652121004824704987274, −5.16864988003492542917016682110, −4.75156976689730144141414214378, −3.13173488171241689632365199273, −2.08853295619630967861878049798, 0.10989499268457192012348188850, 1.73327803801171525311046015014, 3.40986358098600541918385928367, 4.11744870670310615828978331366, 5.10387404152368930720877505970, 5.88118447670809373504231843358, 6.72973512792788119835339305834, 7.51332004808929239143044866838, 8.763860171431102660644184750616, 9.889365433004482073381076051943

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