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Properties

Label 2-1050-35.19-c2-0-24
Degree $2$
Conductor $1050$
Sign $0.952 + 0.304i$
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 + (−6.72 + 1.94i)7-s − 2.82i·8-s + (−1.5 + 2.59i)9-s + (0.263 + 0.455i)11-s + (−1.73 + 2.99i)12-s + 4.22·13-s + (9.61 + 2.37i)14-s + (−2.00 + 3.46i)16-s + (−16.6 − 28.8i)17-s + (3.67 − 2.12i)18-s + (−2.75 − 1.58i)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 + (−0.960 + 0.277i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.0239 + 0.0414i)11-s + (−0.144 + 0.249i)12-s + 0.324·13-s + (0.686 + 0.169i)14-s + (−0.125 + 0.216i)16-s + (−0.980 − 1.69i)17-s + (0.204 − 0.117i)18-s + (−0.144 − 0.0836i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.304i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.952 + 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.952 + 0.304i$
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: \((2,\ 1050,\ (\ :1),\ 0.952 + 0.304i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.200237416\)
\(L(\frac12)\) \(\approx\) \(1.200237416\)
\(L(2)\) 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 \( 1 + (6.72 - 1.94i)T \)
good11 \( 1 + (-0.263 - 0.455i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 4.22T + 169T^{2} \)
17 \( 1 + (16.6 + 28.8i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (2.75 + 1.58i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-9.57 - 5.52i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 56.1T + 841T^{2} \)
31 \( 1 + (1.63 - 0.945i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-8.44 - 4.87i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 4.07iT - 1.68e3T^{2} \)
43 \( 1 + 46.3iT - 1.84e3T^{2} \)
47 \( 1 + (31.6 - 54.7i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-40.2 + 23.2i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-43.7 + 25.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-89.4 - 51.6i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-7.56 + 4.36i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 29.0T + 5.04e3T^{2} \)
73 \( 1 + (8.36 + 14.4i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (66.1 - 114. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 12.4T + 6.88e3T^{2} \)
89 \( 1 + (-59.1 - 34.1i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 149.T + 9.40e3T^{2} \)
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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

−9.676236781125728639606741347358, −8.961279972318675978195691379094, −8.388124747845347475781647442146, −7.13378061247614145310323745674, −6.55417140061673173355663284243, −5.28198590342299365821860182270, −4.23292659518725451595147388976, −3.10038456120361556919242659722, −2.42420902551548187126564657131, −0.62371555882698360860081706496, 0.821037404905924452025408509448, 2.14570233985310514275629583327, 3.33726138597561975020861444665, 4.45468081238490173273657933870, 5.94951795249355541508378826341, 6.51178344643783851420330506661, 7.15549573364430169707707107666, 8.406410336692393575419310769258, 8.590689514228045056110766984125, 9.740309525595513668476077863844

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