Properties

Label 2-1050-35.19-c2-0-4
Degree $2$
Conductor $1050$
Sign $0.313 - 0.949i$
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 + (−1.88 − 6.74i)7-s − 2.82i·8-s + (−1.5 + 2.59i)9-s + (−3 − 5.19i)11-s + (1.73 − 2.99i)12-s − 17.8·13-s + (−2.46 + 9.58i)14-s + (−2.00 + 3.46i)16-s + (−9.37 − 16.2i)17-s + (3.67 − 2.12i)18-s + (14.7 + 8.51i)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 + (−0.268 − 0.963i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.272 − 0.472i)11-s + (0.144 − 0.249i)12-s − 1.37·13-s + (−0.176 + 0.684i)14-s + (−0.125 + 0.216i)16-s + (−0.551 − 0.955i)17-s + (0.204 − 0.117i)18-s + (0.775 + 0.447i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.313 - 0.949i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.313 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.313 - 0.949i$
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: \((2,\ 1050,\ (\ :1),\ 0.313 - 0.949i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1552404828\)
\(L(\frac12)\) \(\approx\) \(0.1552404828\)
\(L(2)\) 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 \( 1 + (1.88 + 6.74i)T \)
good11 \( 1 + (3 + 5.19i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 17.8T + 169T^{2} \)
17 \( 1 + (9.37 + 16.2i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-14.7 - 8.51i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-11.6 - 6.72i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 33.9T + 841T^{2} \)
31 \( 1 + (-12.7 + 7.37i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (5.17 + 2.98i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 35.2iT - 1.68e3T^{2} \)
43 \( 1 + 15.4iT - 1.84e3T^{2} \)
47 \( 1 + (-16.6 + 28.7i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-29.9 + 17.2i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (23.6 - 13.6i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (34.9 + 20.1i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (99.0 - 57.1i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 18.6T + 5.04e3T^{2} \)
73 \( 1 + (-58.5 - 101. i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (44.1 - 76.5i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 75.7T + 6.88e3T^{2} \)
89 \( 1 + (-18 - 10.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 30.5T + 9.40e3T^{2} \)
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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

−9.854375093201783969197700340010, −9.269628081966063527213161121426, −8.090800043207294762443793209108, −7.32180446876295430904858997888, −6.94194143321855771628891148376, −5.65632137423846926537700894443, −4.65230134354830220123584549205, −3.38627102339797634936611315838, −2.36285830344715332842668071881, −0.966127967918405371616556982602, 0.07250510477860099331002743596, 1.97893248775238046285424303892, 3.02676227075413473795041790642, 4.55262277648723554803412645932, 5.30872022134781460642886449265, 6.14793707797283936256385238757, 7.10318628845363099335309184622, 7.894664032445322884653935038826, 9.045298410679327240287387468503, 9.349073563122696947534479091506

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