Properties

Label 2-1120-280.219-c0-0-1
Degree $2$
Conductor $1120$
Sign $0.895 + 0.444i$
Analytic cond. $0.558952$
Root an. cond. $0.747631$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)5-s + 7-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)11-s + 13-s + (−0.5 + 0.866i)19-s + (0.5 − 0.866i)23-s + (−0.499 − 0.866i)25-s + (0.5 − 0.866i)35-s + (−0.5 + 0.866i)37-s − 41-s + (0.499 + 0.866i)45-s + (0.5 − 0.866i)47-s + 49-s + (−0.5 − 0.866i)53-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)5-s + 7-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)11-s + 13-s + (−0.5 + 0.866i)19-s + (0.5 − 0.866i)23-s + (−0.499 − 0.866i)25-s + (0.5 − 0.866i)35-s + (−0.5 + 0.866i)37-s − 41-s + (0.499 + 0.866i)45-s + (0.5 − 0.866i)47-s + 49-s + (−0.5 − 0.866i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $0.895 + 0.444i$
Analytic conductor: \(0.558952\)
Root analytic conductor: \(0.747631\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1120} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1120,\ (\ :0),\ 0.895 + 0.444i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.208792625\)
\(L(\frac12)\) \(\approx\) \(1.208792625\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 - T \)
good3 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - T + T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.16846576731008600779091813970, −8.646081975701094140525373473946, −8.580396719590641342777347651911, −7.84446045348975225843703643558, −6.44518845529571995301613283290, −5.50814084261104868298984074770, −5.03480633452826466468459673362, −3.91875780038000633101148288845, −2.47700450651409867859337629618, −1.35704853684034896538850773240, 1.66369959513530134044079342598, 2.79541741287018713697277611429, 3.86139378875084893608000809995, 5.04240510705274842988192381942, 5.89115990679894339211422985552, 6.78835960436435921430175173794, 7.50577208572087090095483399110, 8.537488976109723831314488477146, 9.255704456255529216337537013479, 10.13391253681635970545600693948

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