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Properties

Label 2-280-280.107-c1-0-20
Degree $2$
Conductor $280$
Sign $-0.191 - 0.981i$
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  + (0.229 + 1.39i)2-s + (3.11 + 0.833i)3-s + (−1.89 + 0.640i)4-s + (1.44 + 1.70i)5-s + (−0.450 + 4.53i)6-s + (−2.23 − 1.42i)7-s + (−1.32 − 2.49i)8-s + (6.39 + 3.68i)9-s + (−2.04 + 2.40i)10-s + (−2.06 − 3.57i)11-s + (−6.43 + 0.411i)12-s + (−1.73 − 1.73i)13-s + (1.47 − 3.43i)14-s + (3.07 + 6.51i)15-s + (3.18 − 2.42i)16-s + (0.356 − 1.32i)17-s + ⋯
L(s)  = 1  + (0.162 + 0.986i)2-s + (1.79 + 0.481i)3-s + (−0.947 + 0.320i)4-s + (0.646 + 0.763i)5-s + (−0.183 + 1.85i)6-s + (−0.842 − 0.538i)7-s + (−0.469 − 0.882i)8-s + (2.13 + 1.22i)9-s + (−0.648 + 0.761i)10-s + (−0.622 − 1.07i)11-s + (−1.85 + 0.118i)12-s + (−0.482 − 0.482i)13-s + (0.394 − 0.918i)14-s + (0.793 + 1.68i)15-s + (0.795 − 0.606i)16-s + (0.0863 − 0.322i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.191 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.191 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.191 - 0.981i$
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: \((2,\ 280,\ (\ :1/2),\ -0.191 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35813 + 1.64909i\)
\(L(\frac12)\) \(\approx\) \(1.35813 + 1.64909i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.229 - 1.39i)T \)
5 \( 1 + (-1.44 - 1.70i)T \)
7 \( 1 + (2.23 + 1.42i)T \)
good3 \( 1 + (-3.11 - 0.833i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (2.06 + 3.57i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.73 + 1.73i)T + 13iT^{2} \)
17 \( 1 + (-0.356 + 1.32i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.51 + 0.872i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.71 + 0.994i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 0.469T + 29T^{2} \)
31 \( 1 + (0.329 - 0.190i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.20 - 4.48i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 8.69T + 41T^{2} \)
43 \( 1 + (5.07 - 5.07i)T - 43iT^{2} \)
47 \( 1 + (-0.284 - 1.06i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.00467 + 0.0174i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (7.49 - 4.32i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.35 - 3.66i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.65 + 9.92i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 11.5iT - 71T^{2} \)
73 \( 1 + (-7.60 - 2.03i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-3.90 + 6.76i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.47 + 2.47i)T - 83iT^{2} \)
89 \( 1 + (-6.00 - 3.46i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.49 + 8.49i)T + 97iT^{2} \)
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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

−12.83295639883716109802837173083, −10.63224206033514733875687068663, −9.904423828503809242248123034342, −9.181302030531750806266486549191, −8.219306841084949192992703812319, −7.37020723550433468433228284462, −6.42688564297949103505021750948, −4.94982994511455900557228954216, −3.44236181033869477292522565706, −2.89666681855111425315946494325, 1.83139202914397365700749729296, 2.58128365403731451441644362119, 3.85821965965105149162027584633, 5.15263142995325332978618486297, 6.81919562715370202707396352373, 8.164389862959783514972787675577, 8.974488905454339572860135151958, 9.609210895865302282063048695721, 10.17862613520478786212852399405, 12.11123389051151338650483504463

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