Properties

Label 2-280-35.17-c1-0-6
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: $\chi_{280} (17, \cdot )$
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

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

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