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Properties

Label 2-1050-35.19-c2-0-38
Degree $2$
Conductor $1050$
Sign $-0.549 + 0.835i$
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 + (5.56 − 4.24i)7-s − 2.82i·8-s + (−1.5 + 2.59i)9-s + (5.42 + 9.40i)11-s + (1.73 − 2.99i)12-s + 0.772·13-s + (−9.81 + 1.26i)14-s + (−2.00 + 3.46i)16-s + (−9.68 − 16.7i)17-s + (3.67 − 2.12i)18-s + (−22.5 − 13.0i)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 + (0.795 − 0.606i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.493 + 0.854i)11-s + (0.144 − 0.249i)12-s + 0.0593·13-s + (−0.701 + 0.0902i)14-s + (−0.125 + 0.216i)16-s + (−0.569 − 0.986i)17-s + (0.204 − 0.117i)18-s + (−1.18 − 0.686i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.549 + 0.835i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.549 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.549 + 0.835i$
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: \((2,\ 1050,\ (\ :1),\ -0.549 + 0.835i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.134832136\)
\(L(\frac12)\) \(\approx\) \(1.134832136\)
\(L(2)\) 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 \( 1 + (-5.56 + 4.24i)T \)
good11 \( 1 + (-5.42 - 9.40i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 0.772T + 169T^{2} \)
17 \( 1 + (9.68 + 16.7i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (22.5 + 13.0i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-11.8 - 6.84i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 6.99T + 841T^{2} \)
31 \( 1 + (-22.7 + 13.1i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-55.9 - 32.3i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 5.54iT - 1.68e3T^{2} \)
43 \( 1 + 68.9iT - 1.84e3T^{2} \)
47 \( 1 + (11.3 - 19.5i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-64.5 + 37.2i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (96.6 - 55.8i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (46.9 + 27.1i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-38.3 + 22.1i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 31.9T + 5.04e3T^{2} \)
73 \( 1 + (53.4 + 92.6i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-14.8 + 25.7i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 15.8T + 6.88e3T^{2} \)
89 \( 1 + (-31.8 - 18.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 134.T + 9.40e3T^{2} \)
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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

−9.417205909880493598934812856835, −8.639437817332047742648946283841, −7.73299490188216135701179420672, −7.06121812650542202655693632168, −6.36306779043521458334216536881, −4.88122049673103198491994037357, −4.24047503345765483026725602769, −2.66091139155145525580915337512, −1.66200185121145812790532052384, −0.50477487356451703674591019512, 1.18456182214321775430968148002, 2.49777231585891312039836230836, 3.97755089043656195593243253703, 4.86415555671841563255466305631, 6.04856575306778067925885200939, 6.32462095975485106745265235406, 7.75698591138300977572061114725, 8.547968377451560178705806749460, 8.926337445814272640079288519337, 9.980556136431084229651391815072

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