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Properties

Label 2-1050-35.19-c2-0-19
Degree $2$
Conductor $1050$
Sign $-0.0196 - 0.999i$
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 + (4.01 + 5.73i)7-s − 2.82i·8-s + (−1.5 + 2.59i)9-s + (8.69 + 15.0i)11-s + (−1.73 + 2.99i)12-s + 7.22·13-s + (−0.854 − 9.86i)14-s + (−2.00 + 3.46i)16-s + (−1.32 − 2.29i)17-s + (3.67 − 2.12i)18-s + (−2.20 − 1.27i)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 + (0.572 + 0.819i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.790 + 1.36i)11-s + (−0.144 + 0.249i)12-s + 0.555·13-s + (−0.0610 − 0.704i)14-s + (−0.125 + 0.216i)16-s + (−0.0779 − 0.135i)17-s + (0.204 − 0.117i)18-s + (−0.115 − 0.0668i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0196 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0196 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.0196 - 0.999i$
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: \((2,\ 1050,\ (\ :1),\ -0.0196 - 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.693023380\)
\(L(\frac12)\) \(\approx\) \(1.693023380\)
\(L(2)\) 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 \( 1 + (-4.01 - 5.73i)T \)
good11 \( 1 + (-8.69 - 15.0i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 7.22T + 169T^{2} \)
17 \( 1 + (1.32 + 2.29i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (2.20 + 1.27i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-34.7 - 20.0i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 47.0T + 841T^{2} \)
31 \( 1 + (-34.9 + 20.1i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (28.1 + 16.2i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 70.6iT - 1.68e3T^{2} \)
43 \( 1 + 37.3iT - 1.84e3T^{2} \)
47 \( 1 + (-16.7 + 28.9i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-61.3 + 35.4i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (87.0 - 50.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (11.1 + 6.44i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-81.4 + 47.0i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 11.5T + 5.04e3T^{2} \)
73 \( 1 + (-11.3 - 19.6i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (12.0 - 20.8i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 111.T + 6.88e3T^{2} \)
89 \( 1 + (-110. - 63.9i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 7.48T + 9.40e3T^{2} \)
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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

−9.641909611334663226251333339937, −9.260558680847978196863539305768, −8.526146553488911672137684831553, −7.60002764053978289075092697170, −6.77437264189546403397020686965, −5.53509686317960982872825579929, −4.58821060912221579237997447256, −3.60149341248562213030892615738, −2.38966388752895675684116434345, −1.44195576272163030709305114247, 0.67987990546191687026275005727, 1.52285524564779037086592091207, 3.07584449534704198196969708607, 4.12391806756942412913789309881, 5.41613889498376114305602480593, 6.40180664107851107412981867851, 7.03439851062010152885997685563, 7.934092187275359433649441971644, 8.688996143723805851774467819756, 9.123538062093299971929064724286

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