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Properties

Label 2-1050-5.2-c2-0-0
Degree $2$
Conductor $1050$
Sign $-0.973 + 0.229i$
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 + i)2-s + (1.22 − 1.22i)3-s + 2i·4-s + 2.44·6-s + (−1.87 − 1.87i)7-s + (−2 + 2i)8-s − 2.99i·9-s − 11.5·11-s + (2.44 + 2.44i)12-s + (7.70 − 7.70i)13-s − 3.74i·14-s − 4·16-s + (−21.0 − 21.0i)17-s + (2.99 − 2.99i)18-s + 24.1i·19-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (0.408 − 0.408i)3-s + 0.5i·4-s + 0.408·6-s + (−0.267 − 0.267i)7-s + (−0.250 + 0.250i)8-s − 0.333i·9-s − 1.04·11-s + (0.204 + 0.204i)12-s + (0.592 − 0.592i)13-s − 0.267i·14-s − 0.250·16-s + (−1.23 − 1.23i)17-s + (0.166 − 0.166i)18-s + 1.26i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.973 + 0.229i$
Analytic conductor: \(28.6104\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1),\ -0.973 + 0.229i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1119522215\)
\(L(\frac12)\) \(\approx\) \(0.1119522215\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
3 \( 1 + (-1.22 + 1.22i)T \)
5 \( 1 \)
7 \( 1 + (1.87 + 1.87i)T \)
good11 \( 1 + 11.5T + 121T^{2} \)
13 \( 1 + (-7.70 + 7.70i)T - 169iT^{2} \)
17 \( 1 + (21.0 + 21.0i)T + 289iT^{2} \)
19 \( 1 - 24.1iT - 361T^{2} \)
23 \( 1 + (30.1 - 30.1i)T - 529iT^{2} \)
29 \( 1 - 51.1iT - 841T^{2} \)
31 \( 1 + 46.9T + 961T^{2} \)
37 \( 1 + (-8.50 - 8.50i)T + 1.36e3iT^{2} \)
41 \( 1 - 18.6T + 1.68e3T^{2} \)
43 \( 1 + (-26.1 + 26.1i)T - 1.84e3iT^{2} \)
47 \( 1 + (50.1 + 50.1i)T + 2.20e3iT^{2} \)
53 \( 1 + (7.08 - 7.08i)T - 2.80e3iT^{2} \)
59 \( 1 + 94.3iT - 3.48e3T^{2} \)
61 \( 1 - 8.09T + 3.72e3T^{2} \)
67 \( 1 + (-20.6 - 20.6i)T + 4.48e3iT^{2} \)
71 \( 1 + 63.7T + 5.04e3T^{2} \)
73 \( 1 + (50.1 - 50.1i)T - 5.32e3iT^{2} \)
79 \( 1 - 1.06iT - 6.24e3T^{2} \)
83 \( 1 + (53.6 - 53.6i)T - 6.88e3iT^{2} \)
89 \( 1 + 145. iT - 7.92e3T^{2} \)
97 \( 1 + (23.7 + 23.7i)T + 9.40e3iT^{2} \)
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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

−10.11580525499200661037480347833, −9.168270011616655549743257513453, −8.292127964224138418931074094353, −7.56677676204808078113444596647, −6.93808967297123090348242214141, −5.84768914505544334325271202615, −5.17987919363511684894139294825, −3.87274629542495600051884146445, −3.10672994928168385236148076412, −1.85295042829759685352359144605, 0.02448201531981650894985927013, 2.04192097203188422850564986526, 2.72885785291529001062973208718, 4.06486864412841365310419294784, 4.52733879581523000883059663922, 5.82973356463341360505331988314, 6.45737067823705976155019238096, 7.75085759080709242356138932536, 8.642954625543819684538152975209, 9.301528997155909358581492591121

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