Properties

Label 2-1050-35.19-c2-0-43
Degree $2$
Conductor $1050$
Sign $-0.263 + 0.964i$
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 + (−5.76 − 3.97i)7-s + 2.82i·8-s + (−1.5 + 2.59i)9-s + (−1.64 − 2.85i)11-s + (−1.73 + 2.99i)12-s − 7.72·13-s + (−4.24 − 8.94i)14-s + (−2.00 + 3.46i)16-s + (−6.30 − 10.9i)17-s + (−3.67 + 2.12i)18-s + (−1.54 − 0.890i)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 + (−0.823 − 0.567i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.149 − 0.259i)11-s + (−0.144 + 0.249i)12-s − 0.594·13-s + (−0.303 − 0.638i)14-s + (−0.125 + 0.216i)16-s + (−0.371 − 0.642i)17-s + (−0.204 + 0.117i)18-s + (−0.0812 − 0.0468i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.263 + 0.964i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.263 + 0.964i$
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: \((2,\ 1050,\ (\ :1),\ -0.263 + 0.964i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6073772129\)
\(L(\frac12)\) \(\approx\) \(0.6073772129\)
\(L(2)\) 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 \( 1 + (5.76 + 3.97i)T \)
good11 \( 1 + (1.64 + 2.85i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 7.72T + 169T^{2} \)
17 \( 1 + (6.30 + 10.9i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (1.54 + 0.890i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (5.85 + 3.37i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 39.2T + 841T^{2} \)
31 \( 1 + (-9.46 + 5.46i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (29.7 + 17.1i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 18.8iT - 1.68e3T^{2} \)
43 \( 1 + 77.4iT - 1.84e3T^{2} \)
47 \( 1 + (6.44 - 11.1i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-44.7 + 25.8i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (97.7 - 56.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (22.7 + 13.1i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (17.2 - 9.95i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 87.4T + 5.04e3T^{2} \)
73 \( 1 + (30.4 + 52.7i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (17.7 - 30.8i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 46.6T + 6.88e3T^{2} \)
89 \( 1 + (-47.4 - 27.4i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 45.7T + 9.40e3T^{2} \)
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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

−9.438868097228276522253269155525, −8.728726781438389339133122003972, −7.56137477166990058522824415042, −7.02146040009309491173935804668, −5.96999791850274211116405308910, −5.09587047659815931859610907064, −4.10681921187096521861003526433, −3.35550441639394411237439042369, −2.29218306631369685791643646717, −0.13169094320852044026516343736, 1.68221088019155216394889385055, 2.66532630278024444160975907387, 3.55097718808011082923191827680, 4.69099683721184077458260267893, 5.76573628517412428397977244999, 6.46140000745652591228624004970, 7.32961905286978846655821876060, 8.296895455442248881718275168307, 9.297479593734469042091741748742, 9.890501709396104604046076912052

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