Properties

Degree $2$
Conductor $1050$
Sign $-0.387 - 0.921i$
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 + (4.22 − 1.77i)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 + (0.921 − 0.387i)21-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(2.002362311\)
\(L(\frac12)\) \(\approx\) \(2.002362311\)
\(L(\frac{3}{2})\) 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

−9.967387146907974064381768134360, −9.342008312214282539744113738176, −8.431497675000231453367696050767, −7.65823471511444656801797511349, −7.13212304164963986819618312753, −5.80037836963358144916291297199, −4.92519658321201342349905784608, −4.02733402092110112665028601119, −3.28357244760207935136789229141, −1.53570434389176955491284794631, 0.925177338972005694605538232554, 2.30483622545207201596114306436, 2.91728346294052171119198843118, 4.11181662973054219296095248027, 5.34889516316225397284979613699, 6.18297487871858723343222126833, 7.39486794251593951140750361101, 8.218453814674227908848543914294, 8.733149434662089782934572661821, 9.635827714905113691992084671367

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