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Properties

Label 2-1120-280.59-c1-0-18
Degree $2$
Conductor $1120$
Sign $-0.522 - 0.852i$
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  + (−0.360 + 0.624i)3-s + (1.56 + 1.59i)5-s + (1.35 + 2.27i)7-s + (1.24 + 2.14i)9-s + (−1.95 + 3.38i)11-s + 2.55i·13-s + (−1.56 + 0.400i)15-s + (2.55 − 4.43i)17-s + (−3.25 + 1.88i)19-s + (−1.90 + 0.0279i)21-s + (−3.50 − 6.06i)23-s + (−0.105 + 4.99i)25-s − 3.94·27-s − 3.39i·29-s + (1.59 − 2.75i)31-s + ⋯
L(s)  = 1  + (−0.208 + 0.360i)3-s + (0.699 + 0.714i)5-s + (0.512 + 0.858i)7-s + (0.413 + 0.716i)9-s + (−0.589 + 1.02i)11-s + 0.708i·13-s + (−0.402 + 0.103i)15-s + (0.620 − 1.07i)17-s + (−0.747 + 0.431i)19-s + (−0.415 + 0.00610i)21-s + (−0.730 − 1.26i)23-s + (−0.0211 + 0.999i)25-s − 0.760·27-s − 0.629i·29-s + (0.286 − 0.495i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $-0.522 - 0.852i$
Analytic conductor: \(8.94324\)
Root analytic conductor: \(2.99052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1120} (1039, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1120,\ (\ :1/2),\ -0.522 - 0.852i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.643730933\)
\(L(\frac12)\) \(\approx\) \(1.643730933\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.56 - 1.59i)T \)
7 \( 1 + (-1.35 - 2.27i)T \)
good3 \( 1 + (0.360 - 0.624i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (1.95 - 3.38i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.55iT - 13T^{2} \)
17 \( 1 + (-2.55 + 4.43i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.25 - 1.88i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.50 + 6.06i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.39iT - 29T^{2} \)
31 \( 1 + (-1.59 + 2.75i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.850 + 1.47i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.97iT - 41T^{2} \)
43 \( 1 + 0.898iT - 43T^{2} \)
47 \( 1 + (-5.64 + 3.25i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.92 + 5.06i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-10.2 - 5.91i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.94 + 10.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.68 - 1.54i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.3iT - 71T^{2} \)
73 \( 1 + (2.59 - 4.49i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.84 + 5.10i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.786T + 83T^{2} \)
89 \( 1 + (0.679 - 0.392i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 16.5T + 97T^{2} \)
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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

−10.02077426153400786247358666372, −9.610638726874352969050022013294, −8.458018853520973906111617043883, −7.60210478387024808788742796620, −6.75633915525898748869349751905, −5.77120314460005855115486511609, −5.00452209914689347667635500652, −4.18682276599998943485954928273, −2.47853570832335440067312227485, −2.04434613250466038687142149982, 0.74971137201475960218974532507, 1.70446181690071806633391661186, 3.33754359001555612616497821324, 4.31453092092987473786110215292, 5.49104260646466562672729751792, 5.99320055951168818299105589514, 7.06889232286726199353350054532, 8.005304093819326934299316824336, 8.594767169725481417466036031581, 9.629460988825061561827766803303

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