Properties

Degree $2$
Conductor $1050$
Sign $-0.921 + 0.387i$
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 + (−1.22 + 1.22i)3-s − 4-s + (1.22 + 1.22i)6-s + (1 − 2.44i)7-s + i·8-s − 2.99i·9-s + (1.22 − 1.22i)12-s + 2.44i·13-s + (−2.44 − i)14-s + 16-s − 4.89·17-s − 2.99·18-s + 2.44i·19-s + (1.77 + 4.22i)21-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.707 + 0.707i)3-s − 0.5·4-s + (0.499 + 0.499i)6-s + (0.377 − 0.925i)7-s + 0.353i·8-s − 0.999i·9-s + (0.353 − 0.353i)12-s + 0.679i·13-s + (−0.654 − 0.267i)14-s + 0.250·16-s − 1.18·17-s − 0.707·18-s + 0.561i·19-s + (0.387 + 0.921i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 + 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.921 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.921 + 0.387i$
Motivic weight: \(1\)
Character: $\chi_{1050} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ -0.921 + 0.387i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5480747325\)
\(L(\frac12)\) \(\approx\) \(0.5480747325\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.22 - 1.22i)T \)
5 \( 1 \)
7 \( 1 + (-1 + 2.44i)T \)
good11 \( 1 - 11T^{2} \)
13 \( 1 - 2.44iT - 13T^{2} \)
17 \( 1 + 4.89T + 17T^{2} \)
19 \( 1 - 2.44iT - 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + 4.89T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + 4.89T + 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 12.2T + 59T^{2} \)
61 \( 1 + 12.2iT - 61T^{2} \)
67 \( 1 + 8T + 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 9.79iT - 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 + 2.44T + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 4.89iT - 97T^{2} \)
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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

−9.852819731215893721979126525747, −8.962877995483475620472544634756, −8.089589908124470474488669096912, −6.84498733593176397322013507869, −6.11871470584013594334326450483, −4.71388457137282644029072364478, −4.41928024822430936023081206773, −3.38507631863004167325127807573, −1.83621883211207400100821098209, −0.27259851216497968689138301265, 1.55682359180407164960799511877, 2.91252501036235362599200584448, 4.58833020679123276816062157146, 5.33583705371475044487665834192, 6.01828880890777802479482610364, 6.89799243615811870020358748348, 7.63779917557935518044545338502, 8.507228337297251442255793586739, 9.163101642626970280542953369970, 10.33011665064657163968570235294

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