Properties

Label 2-280-35.33-c1-0-5
Degree $2$
Conductor $280$
Sign $0.922 - 0.387i$
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  + (0.280 + 0.0752i)3-s + (2.21 + 0.332i)5-s + (1.13 + 2.39i)7-s + (−2.52 − 1.45i)9-s + (1.58 + 2.74i)11-s + (1.12 − 1.12i)13-s + (0.595 + 0.259i)15-s + (0.781 − 2.91i)17-s + (−1.03 + 1.79i)19-s + (0.137 + 0.756i)21-s + (2.21 − 0.593i)23-s + (4.77 + 1.47i)25-s + (−1.21 − 1.21i)27-s + 1.39i·29-s + (0.467 − 0.269i)31-s + ⋯
L(s)  = 1  + (0.162 + 0.0434i)3-s + (0.988 + 0.148i)5-s + (0.427 + 0.904i)7-s + (−0.841 − 0.485i)9-s + (0.477 + 0.826i)11-s + (0.312 − 0.312i)13-s + (0.153 + 0.0670i)15-s + (0.189 − 0.706i)17-s + (−0.237 + 0.411i)19-s + (0.0300 + 0.165i)21-s + (0.461 − 0.123i)23-s + (0.955 + 0.294i)25-s + (−0.233 − 0.233i)27-s + 0.259i·29-s + (0.0839 − 0.0484i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.922 - 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.922 - 0.387i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.922 - 0.387i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53457 + 0.309108i\)
\(L(\frac12)\) \(\approx\) \(1.53457 + 0.309108i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.21 - 0.332i)T \)
7 \( 1 + (-1.13 - 2.39i)T \)
good3 \( 1 + (-0.280 - 0.0752i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (-1.58 - 2.74i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.12 + 1.12i)T - 13iT^{2} \)
17 \( 1 + (-0.781 + 2.91i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.03 - 1.79i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.21 + 0.593i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 1.39iT - 29T^{2} \)
31 \( 1 + (-0.467 + 0.269i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.72 + 6.45i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 1.82iT - 41T^{2} \)
43 \( 1 + (6.50 + 6.50i)T + 43iT^{2} \)
47 \( 1 + (12.4 - 3.32i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.89 + 10.8i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-2.55 - 4.43i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (12.2 + 7.09i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.08 + 0.826i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 7.61T + 71T^{2} \)
73 \( 1 + (-3.08 - 0.827i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (14.7 + 8.52i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.62 + 4.62i)T - 83iT^{2} \)
89 \( 1 + (8.86 - 15.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.71 - 3.71i)T + 97iT^{2} \)
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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

−11.94694908155798860009551307736, −11.03403701313964891888449822324, −9.833076424505313486898411965138, −9.135727391963863548925843126614, −8.297986156465125902623277311342, −6.84585521035114487350298185555, −5.86376137675178577117679302457, −4.97740265732237235804561936891, −3.17789515849469283662211865784, −1.92143101925075073526362094565, 1.50676413050410615482549187809, 3.15397104368441932288979933045, 4.65409387667470370467046673194, 5.82191601036865216994365448303, 6.74730215838553304352665529318, 8.150722985710667582074866095452, 8.823136931913687980056019338237, 9.976689263643922940811796504566, 10.88866741480905128565686324010, 11.56229155838927148259642048029

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