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Properties

Label 2-1120-20.3-c1-0-24
Degree $2$
Conductor $1120$
Sign $0.671 + 0.741i$
Analytic cond. $8.94324$
Root an. cond. $2.99052$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.53 − 1.53i)3-s + (−1.95 + 1.08i)5-s + (0.707 + 0.707i)7-s − 1.71i·9-s − 6.00i·11-s + (4.71 + 4.71i)13-s + (−1.34 + 4.66i)15-s + (−0.423 + 0.423i)17-s + 4.50·19-s + 2.17·21-s + (0.730 − 0.730i)23-s + (2.66 − 4.22i)25-s + (1.97 + 1.97i)27-s − 3.40i·29-s − 8.48i·31-s + ⋯
L(s)  = 1  + (0.886 − 0.886i)3-s + (−0.875 + 0.483i)5-s + (0.267 + 0.267i)7-s − 0.570i·9-s − 1.80i·11-s + (1.30 + 1.30i)13-s + (−0.347 + 1.20i)15-s + (−0.102 + 0.102i)17-s + 1.03·19-s + 0.473·21-s + (0.152 − 0.152i)23-s + (0.533 − 0.845i)25-s + (0.380 + 0.380i)27-s − 0.632i·29-s − 1.52i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 + 0.741i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.671 + 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $0.671 + 0.741i$
Analytic conductor: \(8.94324\)
Root analytic conductor: \(2.99052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1120} (1023, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1120,\ (\ :1/2),\ 0.671 + 0.741i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.073718019\)
\(L(\frac12)\) \(\approx\) \(2.073718019\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (1.95 - 1.08i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (-1.53 + 1.53i)T - 3iT^{2} \)
11 \( 1 + 6.00iT - 11T^{2} \)
13 \( 1 + (-4.71 - 4.71i)T + 13iT^{2} \)
17 \( 1 + (0.423 - 0.423i)T - 17iT^{2} \)
19 \( 1 - 4.50T + 19T^{2} \)
23 \( 1 + (-0.730 + 0.730i)T - 23iT^{2} \)
29 \( 1 + 3.40iT - 29T^{2} \)
31 \( 1 + 8.48iT - 31T^{2} \)
37 \( 1 + (-4.06 + 4.06i)T - 37iT^{2} \)
41 \( 1 + 6.31T + 41T^{2} \)
43 \( 1 + (-0.484 + 0.484i)T - 43iT^{2} \)
47 \( 1 + (-3.48 - 3.48i)T + 47iT^{2} \)
53 \( 1 + (2.52 + 2.52i)T + 53iT^{2} \)
59 \( 1 - 4.52T + 59T^{2} \)
61 \( 1 - 6.61T + 61T^{2} \)
67 \( 1 + (-4.15 - 4.15i)T + 67iT^{2} \)
71 \( 1 - 7.67iT - 71T^{2} \)
73 \( 1 + (3.96 + 3.96i)T + 73iT^{2} \)
79 \( 1 + 13.0T + 79T^{2} \)
83 \( 1 + (0.553 - 0.553i)T - 83iT^{2} \)
89 \( 1 - 12.2iT - 89T^{2} \)
97 \( 1 + (-12.9 + 12.9i)T - 97iT^{2} \)
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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

−9.410054114016791609715759786805, −8.529247068094775156316661422931, −8.243404678314179172909685319686, −7.37565652652980610427526159386, −6.52381224242864683198684650546, −5.71742478447260332853816560959, −4.12021120084151600136935404118, −3.37384529852581959588387806928, −2.40571152262533066507346693093, −1.01999216778232218896870137669, 1.29652286493278055729921236299, 3.03996628956439259283296432829, 3.71816615629013237561710390059, 4.61681759804106258966297405434, 5.28231894086744327152134819915, 6.87361635883791957111612569904, 7.67808347934720532835341208557, 8.405156260150053122335131117744, 9.028164543489258596451915975202, 9.951926356614345184541389560551

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