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Properties

Label 2-1050-35.19-c2-0-1
Degree $2$
Conductor $1050$
Sign $-0.955 + 0.294i$
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 + (−0.264 + 6.99i)7-s + 2.82i·8-s + (−1.5 + 2.59i)9-s + (2.03 + 3.52i)11-s + (1.73 − 2.99i)12-s − 11.1·13-s + (−5.26 + 8.38i)14-s + (−2.00 + 3.46i)16-s + (−1.44 − 2.50i)17-s + (−3.67 + 2.12i)18-s + (−17.9 − 10.3i)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 + (−0.0377 + 0.999i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.184 + 0.320i)11-s + (0.144 − 0.249i)12-s − 0.855·13-s + (−0.376 + 0.598i)14-s + (−0.125 + 0.216i)16-s + (−0.0851 − 0.147i)17-s + (−0.204 + 0.117i)18-s + (−0.944 − 0.545i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 + 0.294i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.955 + 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.955 + 0.294i$
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: \((2,\ 1050,\ (\ :1),\ -0.955 + 0.294i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2778685823\)
\(L(\frac12)\) \(\approx\) \(0.2778685823\)
\(L(2)\) 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 \( 1 + (0.264 - 6.99i)T \)
good11 \( 1 + (-2.03 - 3.52i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 11.1T + 169T^{2} \)
17 \( 1 + (1.44 + 2.50i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (17.9 + 10.3i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (26.5 + 15.3i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 5.75T + 841T^{2} \)
31 \( 1 + (-3.32 + 1.91i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-7.72 - 4.46i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 56.2iT - 1.68e3T^{2} \)
43 \( 1 + 7.16iT - 1.84e3T^{2} \)
47 \( 1 + (23.8 - 41.2i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (28.8 - 16.6i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-38.3 + 22.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (60.3 + 34.8i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (79.7 - 46.0i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 6.64T + 5.04e3T^{2} \)
73 \( 1 + (26.2 + 45.3i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (26.7 - 46.2i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 116.T + 6.88e3T^{2} \)
89 \( 1 + (-85.6 - 49.4i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 132.T + 9.40e3T^{2} \)
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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

−10.21455235087913133162606466294, −9.196300263749415521706973884720, −8.383099703013130752740378882293, −7.53740791734257908657557086475, −6.62660913989626166919269681659, −6.00354641214499374887543813098, −5.06562862492268396659982453443, −4.26018849087006946682810170097, −2.76910504168293560742258362204, −1.99595612464210227782480234922, 0.06611851706266780426154909773, 1.63130975182291730288796342536, 3.06607522958817896772994040992, 4.06165959190501337151226982135, 4.62544225497145577169098014539, 5.75447929372817618671792381919, 6.51763507989770379918663818316, 7.49901226542946273685824411784, 8.440210898558269656112003487466, 9.708741867000045018737789507136

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