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Properties

Label 2-1050-7.3-c2-0-41
Degree $2$
Conductor $1050$
Sign $-0.982 + 0.188i$
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  + (−0.707 − 1.22i)2-s + (−1.5 − 0.866i)3-s + (−0.999 + 1.73i)4-s + 2.44i·6-s + (−0.440 − 6.98i)7-s + 2.82·8-s + (1.5 + 2.59i)9-s + (−3.76 + 6.52i)11-s + (2.99 − 1.73i)12-s − 21.3i·13-s + (−8.24 + 5.47i)14-s + (−2.00 − 3.46i)16-s + (18.1 + 10.4i)17-s + (2.12 − 3.67i)18-s + (20.8 − 12.0i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.5 − 0.288i)3-s + (−0.249 + 0.433i)4-s + 0.408i·6-s + (−0.0628 − 0.998i)7-s + 0.353·8-s + (0.166 + 0.288i)9-s + (−0.342 + 0.593i)11-s + (0.249 − 0.144i)12-s − 1.64i·13-s + (−0.588 + 0.391i)14-s + (−0.125 − 0.216i)16-s + (1.06 + 0.617i)17-s + (0.117 − 0.204i)18-s + (1.09 − 0.634i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.188i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.982 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.982 + 0.188i$
Analytic conductor: \(28.6104\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1),\ -0.982 + 0.188i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9435961424\)
\(L(\frac12)\) \(\approx\) \(0.9435961424\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 1.22i)T \)
3 \( 1 + (1.5 + 0.866i)T \)
5 \( 1 \)
7 \( 1 + (0.440 + 6.98i)T \)
good11 \( 1 + (3.76 - 6.52i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 21.3iT - 169T^{2} \)
17 \( 1 + (-18.1 - 10.4i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-20.8 + 12.0i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-2.79 - 4.83i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 9.96T + 841T^{2} \)
31 \( 1 + (-5.70 - 3.29i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-11.1 - 19.3i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 51.0iT - 1.68e3T^{2} \)
43 \( 1 + 34.7T + 1.84e3T^{2} \)
47 \( 1 + (-67.2 + 38.8i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-24.8 + 43.1i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-72.9 - 42.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (72.4 - 41.8i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-33.2 + 57.6i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 68.1T + 5.04e3T^{2} \)
73 \( 1 + (75.9 + 43.8i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (49.3 + 85.5i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 26.9iT - 6.88e3T^{2} \)
89 \( 1 + (-12.0 + 6.95i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 3.69iT - 9.40e3T^{2} \)
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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

−9.667133190187126340220017592159, −8.425694507288514458132037379928, −7.55123647188090804648484370785, −7.18459181052150521570350925978, −5.75438675576467443481368769395, −5.01690981082561559240532439576, −3.79410702086716083869043106178, −2.87911250966596905999210182094, −1.35698630428371312464232035447, −0.41262566532617654963138977256, 1.28109740176953547811055413450, 2.81737135814270523045139638752, 4.14046358117295050867603799079, 5.27037223085599943407229163050, 5.78126746856201761558295022306, 6.67766490892203049660671110262, 7.60188477105028198104384801212, 8.498384241210215706273322855495, 9.408934532355300320619966951310, 9.736866504605034950460916895321

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