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Properties

Label 2-1050-35.19-c2-0-14
Degree $2$
Conductor $1050$
Sign $0.779 - 0.626i$
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 + (2.33 + 6.59i)7-s − 2.82i·8-s + (−1.5 + 2.59i)9-s + (4.09 + 7.08i)11-s + (1.73 − 2.99i)12-s + 22.1·13-s + (1.80 − 9.73i)14-s + (−2.00 + 3.46i)16-s + (−12.8 − 22.2i)17-s + (3.67 − 2.12i)18-s + (4.69 + 2.70i)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 + (0.333 + 0.942i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.372 + 0.644i)11-s + (0.144 − 0.249i)12-s + 1.70·13-s + (0.129 − 0.695i)14-s + (−0.125 + 0.216i)16-s + (−0.755 − 1.30i)17-s + (0.204 − 0.117i)18-s + (0.247 + 0.142i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.779 - 0.626i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.779 - 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.779 - 0.626i$
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: \((2,\ 1050,\ (\ :1),\ 0.779 - 0.626i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.204422106\)
\(L(\frac12)\) \(\approx\) \(1.204422106\)
\(L(2)\) 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 \( 1 + (-2.33 - 6.59i)T \)
good11 \( 1 + (-4.09 - 7.08i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 22.1T + 169T^{2} \)
17 \( 1 + (12.8 + 22.2i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-4.69 - 2.70i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (1.67 + 0.965i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 11.5T + 841T^{2} \)
31 \( 1 + (52.5 - 30.3i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-53.6 - 30.9i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 78.1iT - 1.68e3T^{2} \)
43 \( 1 + 52.6iT - 1.84e3T^{2} \)
47 \( 1 + (13.6 - 23.5i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (9.84 - 5.68i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-59.0 + 34.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-77.6 - 44.8i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (42.9 - 24.8i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 108.T + 5.04e3T^{2} \)
73 \( 1 + (-6.46 - 11.2i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-29.6 + 51.2i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 52.9T + 6.88e3T^{2} \)
89 \( 1 + (34.4 + 19.8i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 70.3T + 9.40e3T^{2} \)
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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

−9.637896673599412322465743786536, −8.969754183170188621494556834798, −8.327526776643927438556482907559, −7.35999514510485410332029023114, −6.53826226262453582403031154730, −5.67633586115747059215953164852, −4.58545587708472968193395842023, −3.26052452058388064498474940002, −2.11110895192106838308059344413, −1.14915642081076065728714925545, 0.55225918859937350214156776763, 1.73948523056379441225309798428, 3.66764170499162956850977545755, 4.15194018216721589017769470310, 5.62997301589885773244248436478, 6.17993322196174350000469868799, 7.14495964990934827365748767135, 8.126525358903567793994527884012, 8.783108659077481358259208256187, 9.536728448348054697014514940625

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