Properties

Degree $2$
Conductor $1050$
Sign $0.813 - 0.581i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 - 0.581i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.813 - 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.813 - 0.581i$
Motivic weight: \(1\)
Character: $\chi_{1050} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ 0.813 - 0.581i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.288440110\)
\(L(\frac12)\) \(\approx\) \(1.288440110\)
\(L(\frac{3}{2})\) 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

−10.01802329503666478236315690441, −9.284465037851343932643686918731, −8.189442249042407633140789668630, −7.53601587148826521065257996735, −6.59992795123149065499527583032, −5.59647556583183780735154389096, −4.76067425977757417771059968689, −4.19963938735050030364786509164, −2.92660031237026692101455370763, −0.74284526345787270661054444972, 1.18810336651684190208766638938, 2.16291686930199594754261515317, 3.29533244862797346730239755978, 4.59980135269796331023497319651, 5.77702935507217244449358446306, 6.09279234210268163352355130204, 7.55837433340425033195602379336, 8.145462861150052695384908516053, 9.043294422479205197707139907337, 9.787734368485289738026898530599

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