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Properties

Label 2-280-40.29-c1-0-16
Degree $2$
Conductor $280$
Sign $0.864 - 0.502i$
Analytic cond. $2.23581$
Root an. cond. $1.49526$
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  + (−1.12 + 0.856i)2-s + 1.65·3-s + (0.534 − 1.92i)4-s + (2.21 − 0.327i)5-s + (−1.86 + 1.41i)6-s + i·7-s + (1.04 + 2.62i)8-s − 0.266·9-s + (−2.20 + 2.26i)10-s − 0.550i·11-s + (0.883 − 3.18i)12-s + 3.69·13-s + (−0.856 − 1.12i)14-s + (3.65 − 0.541i)15-s + (−3.42 − 2.06i)16-s − 2.20i·17-s + ⋯
L(s)  = 1  + (−0.795 + 0.605i)2-s + 0.954·3-s + (0.267 − 0.963i)4-s + (0.989 − 0.146i)5-s + (−0.759 + 0.577i)6-s + 0.377i·7-s + (0.370 + 0.928i)8-s − 0.0888·9-s + (−0.698 + 0.715i)10-s − 0.165i·11-s + (0.255 − 0.919i)12-s + 1.02·13-s + (−0.228 − 0.300i)14-s + (0.944 − 0.139i)15-s + (−0.857 − 0.515i)16-s − 0.534i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 - 0.502i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.864 - 0.502i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.864 - 0.502i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.864 - 0.502i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31435 + 0.354329i\)
\(L(\frac12)\) \(\approx\) \(1.31435 + 0.354329i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.12 - 0.856i)T \)
5 \( 1 + (-2.21 + 0.327i)T \)
7 \( 1 - iT \)
good3 \( 1 - 1.65T + 3T^{2} \)
11 \( 1 + 0.550iT - 11T^{2} \)
13 \( 1 - 3.69T + 13T^{2} \)
17 \( 1 + 2.20iT - 17T^{2} \)
19 \( 1 + 4.02iT - 19T^{2} \)
23 \( 1 - 7.96iT - 23T^{2} \)
29 \( 1 - 2.33iT - 29T^{2} \)
31 \( 1 - 1.60T + 31T^{2} \)
37 \( 1 + 7.31T + 37T^{2} \)
41 \( 1 + 0.475T + 41T^{2} \)
43 \( 1 + 6.94T + 43T^{2} \)
47 \( 1 - 4.81iT - 47T^{2} \)
53 \( 1 - 9.39T + 53T^{2} \)
59 \( 1 - 1.71iT - 59T^{2} \)
61 \( 1 + 13.9iT - 61T^{2} \)
67 \( 1 + 8.80T + 67T^{2} \)
71 \( 1 + 11.7T + 71T^{2} \)
73 \( 1 + 14.2iT - 73T^{2} \)
79 \( 1 + 7.79T + 79T^{2} \)
83 \( 1 + 4.75T + 83T^{2} \)
89 \( 1 + 4.88T + 89T^{2} \)
97 \( 1 - 18.8iT - 97T^{2} \)
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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

−11.70285313615541505211669747376, −10.73049966550112651963105623866, −9.594684981405051377565741978206, −9.013339174788959453307041005181, −8.385110595108761871477315017652, −7.19261042753964290746357625512, −6.05926175320053777788398502056, −5.17170785183363565966156740265, −3.06046546084892109753461267373, −1.69535850809491821798655379948, 1.67741083474657735759961450546, 2.86091899456474242412423312056, 4.01726820654956503143082245122, 5.98796409665597674621784222649, 7.11711861494070871544000976021, 8.526409987051535593240147432809, 8.666971646795848389575749445590, 10.06925948223086465810493505256, 10.40326881833081249632340081203, 11.62137156625724836132571266961

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