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Properties

Label 2-1050-35.24-c2-0-43
Degree $2$
Conductor $1050$
Sign $-0.968 + 0.247i$
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 + (−2.33 + 6.59i)7-s − 2.82i·8-s + (−1.5 − 2.59i)9-s + (4.09 − 7.08i)11-s + (−1.73 − 2.99i)12-s − 22.1·13-s + (1.80 + 9.73i)14-s + (−2.00 − 3.46i)16-s + (12.8 − 22.2i)17-s + (−3.67 − 2.12i)18-s + (4.69 − 2.70i)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 + (−0.333 + 0.942i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + (0.372 − 0.644i)11-s + (−0.144 − 0.249i)12-s − 1.70·13-s + (0.129 + 0.695i)14-s + (−0.125 − 0.216i)16-s + (0.755 − 1.30i)17-s + (−0.204 − 0.117i)18-s + (0.247 − 0.142i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.247i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.968 + 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.968 + 0.247i$
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: \((2,\ 1050,\ (\ :1),\ -0.968 + 0.247i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.651537041\)
\(L(\frac12)\) \(\approx\) \(1.651537041\)
\(L(2)\) 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 \( 1 + (2.33 - 6.59i)T \)
good11 \( 1 + (-4.09 + 7.08i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 22.1T + 169T^{2} \)
17 \( 1 + (-12.8 + 22.2i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-4.69 + 2.70i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-1.67 + 0.965i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 11.5T + 841T^{2} \)
31 \( 1 + (52.5 + 30.3i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (53.6 - 30.9i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 78.1iT - 1.68e3T^{2} \)
43 \( 1 + 52.6iT - 1.84e3T^{2} \)
47 \( 1 + (-13.6 - 23.5i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-9.84 - 5.68i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-59.0 - 34.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-77.6 + 44.8i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-42.9 - 24.8i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 108.T + 5.04e3T^{2} \)
73 \( 1 + (6.46 - 11.2i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-29.6 - 51.2i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 52.9T + 6.88e3T^{2} \)
89 \( 1 + (34.4 - 19.8i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 70.3T + 9.40e3T^{2} \)
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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

−9.378186464505362364379311401988, −8.674126680775675793256092004881, −7.37982527372785115329848755321, −6.94322951051389967504008176386, −5.55064486334926947350047728092, −5.29170070878196022436061919424, −3.76207773622926778417068151648, −2.81717541416069936302929208032, −2.04828665032343441487765363409, −0.36343908422683506963542367122, 1.77875035766479197371417900133, 3.19323112820505556815266415071, 3.96085028216023921329077797646, 4.81270576187944242113790754620, 5.66976722591320128894672820923, 6.93067048150616388871969657148, 7.37696881919634840957813681672, 8.291090150973215946263517430758, 9.458019607345987924087463287331, 10.03785927871034953952981989429

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