Properties

Label 2-1050-35.19-c2-0-34
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

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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 + (6.50 + 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: $\chi_{1050} (649, \cdot )$
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 \( 1 - 49.4T + 9.40e3T^{2} \)
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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

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

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