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Properties

Label 2-280-280.237-c1-0-43
Degree $2$
Conductor $280$
Sign $-0.957 + 0.289i$
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  + (0.255 − 1.39i)2-s + (1.92 − 1.92i)3-s + (−1.86 − 0.712i)4-s + (−2.22 + 0.200i)5-s + (−2.18 − 3.16i)6-s + (−0.516 − 2.59i)7-s + (−1.46 + 2.41i)8-s − 4.40i·9-s + (−0.291 + 3.14i)10-s + 5.44i·11-s + (−4.96 + 2.22i)12-s + (1.42 − 1.42i)13-s + (−3.74 + 0.0540i)14-s + (−3.90 + 4.67i)15-s + (2.98 + 2.66i)16-s + (4.92 − 4.92i)17-s + ⋯
L(s)  = 1  + (0.180 − 0.983i)2-s + (1.11 − 1.11i)3-s + (−0.934 − 0.356i)4-s + (−0.995 + 0.0895i)5-s + (−0.891 − 1.29i)6-s + (−0.195 − 0.980i)7-s + (−0.519 + 0.854i)8-s − 1.46i·9-s + (−0.0922 + 0.995i)10-s + 1.64i·11-s + (−1.43 + 0.642i)12-s + (0.395 − 0.395i)13-s + (−0.999 + 0.0144i)14-s + (−1.00 + 1.20i)15-s + (0.746 + 0.665i)16-s + (1.19 − 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.957 + 0.289i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.957 + 0.289i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.215186 - 1.45494i\)
\(L(\frac12)\) \(\approx\) \(0.215186 - 1.45494i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.255 + 1.39i)T \)
5 \( 1 + (2.22 - 0.200i)T \)
7 \( 1 + (0.516 + 2.59i)T \)
good3 \( 1 + (-1.92 + 1.92i)T - 3iT^{2} \)
11 \( 1 - 5.44iT - 11T^{2} \)
13 \( 1 + (-1.42 + 1.42i)T - 13iT^{2} \)
17 \( 1 + (-4.92 + 4.92i)T - 17iT^{2} \)
19 \( 1 + 3.36iT - 19T^{2} \)
23 \( 1 + (-1.59 + 1.59i)T - 23iT^{2} \)
29 \( 1 + 2.04T + 29T^{2} \)
31 \( 1 - 1.78iT - 31T^{2} \)
37 \( 1 + (1.63 - 1.63i)T - 37iT^{2} \)
41 \( 1 - 1.17iT - 41T^{2} \)
43 \( 1 + (5.08 + 5.08i)T + 43iT^{2} \)
47 \( 1 + (1.67 - 1.67i)T - 47iT^{2} \)
53 \( 1 + (-8.44 - 8.44i)T + 53iT^{2} \)
59 \( 1 - 8.44iT - 59T^{2} \)
61 \( 1 - 3.37T + 61T^{2} \)
67 \( 1 + (5.38 - 5.38i)T - 67iT^{2} \)
71 \( 1 - 10.6T + 71T^{2} \)
73 \( 1 + (-2.52 - 2.52i)T + 73iT^{2} \)
79 \( 1 - 5.29iT - 79T^{2} \)
83 \( 1 + (0.118 - 0.118i)T - 83iT^{2} \)
89 \( 1 - 0.578T + 89T^{2} \)
97 \( 1 + (-1.57 + 1.57i)T - 97iT^{2} \)
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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

−11.73221341860111187824877259599, −10.52017503211743870375064423061, −9.582726769001225595714736467592, −8.524389822104471064450747836399, −7.49242370719777830464186452241, −7.03986793471637750386561011235, −4.82581947126045779394220130879, −3.65149834050434482391708029307, −2.67118155375199095044419774751, −1.05483879993791642173933915879, 3.42997810224932484586988914977, 3.66871864495469598232033762729, 5.18702945994698444644457597455, 6.19915030081933529227279494389, 7.929350316139929629368639217858, 8.409647013005990023461644860697, 9.014114983029514959798470058847, 10.06204181358239565601331626174, 11.33625847482537713345509431719, 12.44659478398302648921108261784

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