Properties

Label 2-1050-35.24-c2-0-20
Degree $2$
Conductor $1050$
Sign $0.999 - 0.0426i$
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

Related objects

Downloads

Learn more

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 + (6.50 − 2.59i)7-s − 2.82i·8-s + (−1.5 − 2.59i)9-s + (−5.13 + 8.89i)11-s + (1.73 + 2.99i)12-s + 7.02·13-s + (6.12 − 7.77i)14-s + (−2.00 − 3.46i)16-s + (−15.8 + 27.4i)17-s + (−3.67 − 2.12i)18-s + (26.9 − 15.5i)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 + (0.928 − 0.370i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + (−0.466 + 0.808i)11-s + (0.144 + 0.249i)12-s + 0.540·13-s + (0.437 − 0.555i)14-s + (−0.125 − 0.216i)16-s + (−0.933 + 1.61i)17-s + (−0.204 − 0.117i)18-s + (1.41 − 0.818i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0426i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.999 - 0.0426i$
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: \((2,\ 1050,\ (\ :1),\ 0.999 - 0.0426i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.843628084\)
\(L(\frac12)\) \(\approx\) \(2.843628084\)
\(L(2)\) 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 \( 1 + (-6.50 + 2.59i)T \)
good11 \( 1 + (5.13 - 8.89i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 7.02T + 169T^{2} \)
17 \( 1 + (15.8 - 27.4i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-26.9 + 15.5i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-20.5 + 11.8i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 9.19T + 841T^{2} \)
31 \( 1 + (-17.4 - 10.0i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-41.7 + 24.0i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 65.1iT - 1.68e3T^{2} \)
43 \( 1 - 3.03iT - 1.84e3T^{2} \)
47 \( 1 + (30.9 + 53.6i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-1.19 - 0.690i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-95.1 - 54.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (34.3 - 19.8i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (13.7 + 7.95i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 53.3T + 5.04e3T^{2} \)
73 \( 1 + (-36.1 + 62.6i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-53.2 - 92.1i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 49.4T + 6.88e3T^{2} \)
89 \( 1 + (-142. + 82.4i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 49.4T + 9.40e3T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.990049430393583242570278761569, −9.002759958054465131692731587628, −8.080472392952853703563687261838, −7.08993252790928918840529354264, −6.16686260489583924747980107919, −5.07711574740589180536181517076, −4.57448712204094930092204729515, −3.64122758236327837704598094080, −2.35127421821968101880134628957, −1.09472284910393515100581184859, 0.940063855608872871027050384533, 2.37654753012838558691638159540, 3.40197396730168327602196337140, 4.80412760969583938542411951833, 5.37510200616745385823519970402, 6.17961139937172125459779941206, 7.24036421655662467443150401622, 7.84703645497973160086404238727, 8.669254376609133981358479351694, 9.588349562461472016616347037600

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