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Properties

Label 2-1120-280.13-c1-0-5
Degree $2$
Conductor $1120$
Sign $0.904 + 0.425i$
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  + (−1.96 − 1.96i)3-s + (−0.254 − 2.22i)5-s + (−1.87 + 1.87i)7-s + 4.74i·9-s + (4.88 + 4.88i)13-s + (−3.87 + 4.87i)15-s − 2.41i·19-s + 7.36·21-s + (6.74 + 6.74i)23-s + (−4.87 + 1.12i)25-s + (3.42 − 3.42i)27-s + (4.63 + 3.68i)35-s − 19.2i·39-s + (10.5 − 1.20i)45-s − 7i·49-s + ⋯
L(s)  = 1  + (−1.13 − 1.13i)3-s + (−0.113 − 0.993i)5-s + (−0.707 + 0.707i)7-s + 1.58i·9-s + (1.35 + 1.35i)13-s + (−0.999 + 1.25i)15-s − 0.552i·19-s + 1.60·21-s + (1.40 + 1.40i)23-s + (−0.974 + 0.225i)25-s + (0.659 − 0.659i)27-s + (0.782 + 0.622i)35-s − 3.07i·39-s + (1.57 − 0.179i)45-s − i·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.425i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.904 + 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $0.904 + 0.425i$
Analytic conductor: \(8.94324\)
Root analytic conductor: \(2.99052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1120} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1120,\ (\ :1/2),\ 0.904 + 0.425i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9054362394\)
\(L(\frac12)\) \(\approx\) \(0.9054362394\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.254 + 2.22i)T \)
7 \( 1 + (1.87 - 1.87i)T \)
good3 \( 1 + (1.96 + 1.96i)T + 3iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (-4.88 - 4.88i)T + 13iT^{2} \)
17 \( 1 + 17iT^{2} \)
19 \( 1 + 2.41iT - 19T^{2} \)
23 \( 1 + (-6.74 - 6.74i)T + 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 - 53iT^{2} \)
59 \( 1 + 10.4iT - 59T^{2} \)
61 \( 1 + 6.47T + 61T^{2} \)
67 \( 1 + 67iT^{2} \)
71 \( 1 - 15.2T + 71T^{2} \)
73 \( 1 - 73iT^{2} \)
79 \( 1 - 8.25iT - 79T^{2} \)
83 \( 1 + (-12.2 - 12.2i)T + 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 97iT^{2} \)
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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

−9.487516992908963015819277078799, −9.028697918944278529368078067305, −8.098106057667941735027784395267, −7.00444317041011216256185486951, −6.42806103246830757796047201558, −5.63398283866769738331705109735, −4.91539054181101343907412625580, −3.59500492625356168714462192381, −1.91854399737189503265027566937, −0.945198921250729642269508848638, 0.63894087885402324468949746112, 3.06170289917278022764448738124, 3.69318461632874718489411016173, 4.65192489990083232930021520477, 5.80728484600472618083138905844, 6.28820476187032781405047901703, 7.16359945136244566953818988247, 8.279171749074187156436443119120, 9.395107387571480495005831997334, 10.28537004483072026962761949412

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