Properties

Degree $2$
Conductor $1050$
Sign $-0.0899 + 0.995i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s + (−0.721 − 1.57i)3-s − 4-s + (1.57 − 0.721i)6-s + (−1.31 + 2.29i)7-s i·8-s + (−1.95 + 2.27i)9-s − 3.91i·11-s + (0.721 + 1.57i)12-s + 4.99i·13-s + (−2.29 − 1.31i)14-s + 16-s + 3.54·17-s + (−2.27 − 1.95i)18-s − 3.14i·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.416 − 0.909i)3-s − 0.5·4-s + (0.642 − 0.294i)6-s + (−0.496 + 0.867i)7-s − 0.353i·8-s + (−0.652 + 0.757i)9-s − 1.18i·11-s + (0.208 + 0.454i)12-s + 1.38i·13-s + (−0.613 − 0.351i)14-s + 0.250·16-s + 0.860·17-s + (−0.535 − 0.461i)18-s − 0.722i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0899 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0899 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.0899 + 0.995i$
Motivic weight: \(1\)
Character: $\chi_{1050} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ -0.0899 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6393145263\)
\(L(\frac12)\) \(\approx\) \(0.6393145263\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (0.721 + 1.57i)T \)
5 \( 1 \)
7 \( 1 + (1.31 - 2.29i)T \)
good11 \( 1 + 3.91iT - 11T^{2} \)
13 \( 1 - 4.99iT - 13T^{2} \)
17 \( 1 - 3.54T + 17T^{2} \)
19 \( 1 + 3.14iT - 19T^{2} \)
23 \( 1 + 7.54iT - 23T^{2} \)
29 \( 1 + 7.54iT - 29T^{2} \)
31 \( 1 + 4.19iT - 31T^{2} \)
37 \( 1 + 10.4T + 37T^{2} \)
41 \( 1 + 9.32T + 41T^{2} \)
43 \( 1 + 2.91T + 43T^{2} \)
47 \( 1 + 8.00T + 47T^{2} \)
53 \( 1 - 0.288iT - 53T^{2} \)
59 \( 1 - 5.89T + 59T^{2} \)
61 \( 1 + 2.48iT - 61T^{2} \)
67 \( 1 - 0.545T + 67T^{2} \)
71 \( 1 + 5.37iT - 71T^{2} \)
73 \( 1 + 4.85iT - 73T^{2} \)
79 \( 1 - 0.742T + 79T^{2} \)
83 \( 1 - 4.45T + 83T^{2} \)
89 \( 1 - 12.3T + 89T^{2} \)
97 \( 1 + 0.524iT - 97T^{2} \)
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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

−9.436532464307206658768004995540, −8.595250201391146561405826523941, −8.124173830747916729141474796253, −6.86164223992845623724128355250, −6.43416939953119687213207811054, −5.68415124167787086399133003714, −4.77361322643616952773953042359, −3.30930975316153323217875783456, −2.06829389242040314291249005723, −0.31753769077783261438014609348, 1.38924976510596922884669902560, 3.35971592558128000491386456495, 3.56137376964583380254074398501, 4.96903305048187364115507317282, 5.42755859379224433806433804616, 6.77789567177742632584997640483, 7.69035950343014821531485196653, 8.732132119220194522333556258798, 9.829417270946106511454061581481, 10.12910443499150900951828097992

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