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Properties

Label 2-1050-35.24-c2-0-19
Degree $2$
Conductor $1050$
Sign $0.930 + 0.367i$
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 + (0.647 − 1.12i)11-s + (1.73 + 2.99i)12-s + 3.22·13-s + (−4.24 + 8.94i)14-s + (−2.00 − 3.46i)16-s + (15.1 − 26.2i)17-s + (−3.67 − 2.12i)18-s + (−28.9 + 16.7i)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 + (0.0588 − 0.101i)11-s + (0.144 + 0.249i)12-s + 0.248·13-s + (−0.303 + 0.638i)14-s + (−0.125 − 0.216i)16-s + (0.893 − 1.54i)17-s + (−0.204 − 0.117i)18-s + (−1.52 + 0.879i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 + 0.367i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.930 + 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.930 + 0.367i$
Analytic conductor: \(28.6104\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1),\ 0.930 + 0.367i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.242419454\)
\(L(\frac12)\) \(\approx\) \(2.242419454\)
\(L(2)\) 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 \( 1 + (-0.647 + 1.12i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 3.22T + 169T^{2} \)
17 \( 1 + (-15.1 + 26.2i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (28.9 - 16.7i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-30.9 + 17.8i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 32.8T + 841T^{2} \)
31 \( 1 + (-39.5 - 22.8i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-34.6 + 20.0i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 42.2iT - 1.68e3T^{2} \)
43 \( 1 - 55.4iT - 1.84e3T^{2} \)
47 \( 1 + (30.3 + 52.6i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-47.3 - 27.3i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (11.4 + 6.59i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-34.1 + 19.7i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-34.7 - 20.0i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 46.4T + 5.04e3T^{2} \)
73 \( 1 + (-68.2 + 118. i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (21.1 + 36.6i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 1.53T + 6.88e3T^{2} \)
89 \( 1 + (-30.2 + 17.4i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 150.T + 9.40e3T^{2} \)
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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

−9.864852793404063452932633466663, −9.047665723682331631234526407688, −8.145379045621024456362020205460, −6.70528866462355686660734970301, −6.25194797210112375905234042531, −5.19723910368356167887024017933, −4.47320272618254603635038028281, −3.27909376930891161705730467674, −2.60094313780949988914857884049, −0.789546404636446001727360630459, 0.937998066113458779694047016295, 2.49921578844200502860215506583, 3.63045250231413156788072006773, 4.50067751368699761941104833621, 5.66089442534911303368670141007, 6.47175860020781512434452377807, 6.93165902215166465753951594330, 7.989222987535708054706218153346, 8.703363378235741666492768758107, 9.910242644235566914367976111482

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