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Properties

Label 2-1050-35.19-c2-0-40
Degree $2$
Conductor $1050$
Sign $-0.745 + 0.665i$
Analytic cond. $28.6104$
Root an. cond. $5.34887$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 + 0.707i)2-s + (−0.866 − 1.5i)3-s + (0.999 + 1.73i)4-s − 2.44i·6-s + (−6.46 + 2.67i)7-s + 2.82i·8-s + (−1.5 + 2.59i)9-s + (−0.578 − 1.00i)11-s + (1.73 − 2.99i)12-s + 14.8·13-s + (−9.81 − 1.29i)14-s + (−2.00 + 3.46i)16-s + (−6.30 − 10.9i)17-s + (−3.67 + 2.12i)18-s + (−16.7 − 9.65i)19-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.288 − 0.5i)3-s + (0.249 + 0.433i)4-s − 0.408i·6-s + (−0.924 + 0.381i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.0525 − 0.0910i)11-s + (0.144 − 0.249i)12-s + 1.13·13-s + (−0.700 − 0.0928i)14-s + (−0.125 + 0.216i)16-s + (−0.371 − 0.642i)17-s + (−0.204 + 0.117i)18-s + (−0.879 − 0.507i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.745 + 0.665i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.745 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.745 + 0.665i$
Analytic conductor: \(28.6104\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1),\ -0.745 + 0.665i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4856711938\)
\(L(\frac12)\) \(\approx\) \(0.4856711938\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 - 0.707i)T \)
3 \( 1 + (0.866 + 1.5i)T \)
5 \( 1 \)
7 \( 1 + (6.46 - 2.67i)T \)
good11 \( 1 + (0.578 + 1.00i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 14.8T + 169T^{2} \)
17 \( 1 + (6.30 + 10.9i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (16.7 + 9.65i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-21.0 - 12.1i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 49.0T + 841T^{2} \)
31 \( 1 + (24.9 - 14.4i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (46.1 + 26.6i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 38.0iT - 1.68e3T^{2} \)
43 \( 1 + 63.5iT - 1.84e3T^{2} \)
47 \( 1 + (-12.5 + 21.8i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-18.0 + 10.4i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (21.1 - 12.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-5.53 - 3.19i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (107. - 62.2i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 118.T + 5.04e3T^{2} \)
73 \( 1 + (19.7 + 34.2i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (46.4 - 80.4i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 5.79T + 6.88e3T^{2} \)
89 \( 1 + (131. + 75.9i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 144.T + 9.40e3T^{2} \)
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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

−8.933302995947672080745036821781, −8.842077402596345659426090615048, −7.24002759553882339068720159293, −6.96960224770522147758072453140, −5.85105749519134572828894328461, −5.41402221886348693928755944864, −4.01887672286189656157415091488, −3.14411316240573073214175518790, −1.92129817356645248464044138327, −0.11973633570732641634645600774, 1.54034399315263090556638474804, 3.07509276990440079816047092104, 3.84096683957432677566476835656, 4.60850928676239248297421191640, 5.89921236439237487831097142351, 6.29085801208472719286536042502, 7.33667401423412291958488330156, 8.611141992604834198302236504784, 9.350547592738304167236271296839, 10.27773307431011598399222834646

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