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Properties

Label 2-280-280.107-c1-0-3
Degree $2$
Conductor $280$
Sign $-0.382 - 0.923i$
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  + (−1.25 + 0.653i)2-s + (−0.862 − 0.231i)3-s + (1.14 − 1.63i)4-s + (−1.04 − 1.97i)5-s + (1.23 − 0.273i)6-s + (−2.36 + 1.19i)7-s + (−0.367 + 2.80i)8-s + (−1.90 − 1.10i)9-s + (2.60 + 1.79i)10-s + (2.58 + 4.47i)11-s + (−1.36 + 1.14i)12-s + (2.64 + 2.64i)13-s + (2.18 − 3.03i)14-s + (0.443 + 1.94i)15-s + (−1.37 − 3.75i)16-s + (−0.725 + 2.70i)17-s + ⋯
L(s)  = 1  + (−0.886 + 0.461i)2-s + (−0.497 − 0.133i)3-s + (0.573 − 0.819i)4-s + (−0.466 − 0.884i)5-s + (0.503 − 0.111i)6-s + (−0.892 + 0.450i)7-s + (−0.129 + 0.991i)8-s + (−0.636 − 0.367i)9-s + (0.822 + 0.568i)10-s + (0.779 + 1.35i)11-s + (−0.394 + 0.331i)12-s + (0.732 + 0.732i)13-s + (0.583 − 0.812i)14-s + (0.114 + 0.502i)15-s + (−0.342 − 0.939i)16-s + (−0.175 + 0.656i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.382 - 0.923i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.382 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.212541 + 0.318156i\)
\(L(\frac12)\) \(\approx\) \(0.212541 + 0.318156i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.25 - 0.653i)T \)
5 \( 1 + (1.04 + 1.97i)T \)
7 \( 1 + (2.36 - 1.19i)T \)
good3 \( 1 + (0.862 + 0.231i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (-2.58 - 4.47i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.64 - 2.64i)T + 13iT^{2} \)
17 \( 1 + (0.725 - 2.70i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-5.82 - 3.36i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.16 - 1.38i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 8.59T + 29T^{2} \)
31 \( 1 + (-1.82 + 1.05i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.450 + 1.67i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 2.04T + 41T^{2} \)
43 \( 1 + (3.38 - 3.38i)T - 43iT^{2} \)
47 \( 1 + (-0.488 - 1.82i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.759 + 2.83i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (5.92 - 3.41i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.25 - 3.61i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.41 + 8.99i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 11.8iT - 71T^{2} \)
73 \( 1 + (-2.45 - 0.657i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (5.08 - 8.81i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.87 - 1.87i)T - 83iT^{2} \)
89 \( 1 + (12.9 + 7.48i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.762 - 0.762i)T + 97iT^{2} \)
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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

−11.90302597914215894579831916942, −11.43464852542905223755499136333, −9.845677201118242276686964355415, −9.328264416957499650819984684021, −8.445647437605552031002214903852, −7.28759028074991614987350468800, −6.26204898594013754328080565706, −5.48506157486284480405582244759, −3.86236291815309295809990724868, −1.58835306284410342451487447939, 0.41656831766593446167949320084, 2.97855867894036628650100066344, 3.66382691785293143880973647670, 5.84702420857802393152696024550, 6.72511473482579849836805206865, 7.76203058961074969982701790740, 8.778097873609510150419854533388, 9.848069255684406462973100847484, 10.77956481714762162148819766401, 11.33591412180359883215622793810

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