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Properties

Label 2-280-8.3-c2-0-37
Degree $2$
Conductor $280$
Sign $-0.742 + 0.670i$
Analytic cond. $7.62944$
Root an. cond. $2.76214$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.94 − 0.484i)2-s − 4.98·3-s + (3.53 − 1.88i)4-s + 2.23i·5-s + (−9.67 + 2.41i)6-s − 2.64i·7-s + (5.93 − 5.36i)8-s + 15.8·9-s + (1.08 + 4.33i)10-s − 12.9·11-s + (−17.6 + 9.38i)12-s − 18.4i·13-s + (−1.28 − 5.13i)14-s − 11.1i·15-s + (8.92 − 13.2i)16-s − 10.9·17-s + ⋯
L(s)  = 1  + (0.970 − 0.242i)2-s − 1.66·3-s + (0.882 − 0.470i)4-s + 0.447i·5-s + (−1.61 + 0.402i)6-s − 0.377i·7-s + (0.742 − 0.670i)8-s + 1.76·9-s + (0.108 + 0.433i)10-s − 1.17·11-s + (−1.46 + 0.781i)12-s − 1.41i·13-s + (−0.0915 − 0.366i)14-s − 0.743i·15-s + (0.557 − 0.829i)16-s − 0.647·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.742 + 0.670i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.742 + 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.742 + 0.670i$
Analytic conductor: \(7.62944\)
Root analytic conductor: \(2.76214\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1),\ -0.742 + 0.670i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.344557 - 0.895953i\)
\(L(\frac12)\) \(\approx\) \(0.344557 - 0.895953i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.94 + 0.484i)T \)
5 \( 1 - 2.23iT \)
7 \( 1 + 2.64iT \)
good3 \( 1 + 4.98T + 9T^{2} \)
11 \( 1 + 12.9T + 121T^{2} \)
13 \( 1 + 18.4iT - 169T^{2} \)
17 \( 1 + 10.9T + 289T^{2} \)
19 \( 1 + 32.5T + 361T^{2} \)
23 \( 1 + 33.8iT - 529T^{2} \)
29 \( 1 + 31.8iT - 841T^{2} \)
31 \( 1 - 31.5iT - 961T^{2} \)
37 \( 1 - 26.8iT - 1.36e3T^{2} \)
41 \( 1 - 18.8T + 1.68e3T^{2} \)
43 \( 1 - 41.2T + 1.84e3T^{2} \)
47 \( 1 + 61.2iT - 2.20e3T^{2} \)
53 \( 1 - 32.0iT - 2.80e3T^{2} \)
59 \( 1 + 62.4T + 3.48e3T^{2} \)
61 \( 1 - 28.0iT - 3.72e3T^{2} \)
67 \( 1 - 5.19T + 4.48e3T^{2} \)
71 \( 1 + 36.1iT - 5.04e3T^{2} \)
73 \( 1 - 103.T + 5.32e3T^{2} \)
79 \( 1 - 138. iT - 6.24e3T^{2} \)
83 \( 1 - 61.0T + 6.88e3T^{2} \)
89 \( 1 - 77.4T + 7.92e3T^{2} \)
97 \( 1 + 59.9T + 9.40e3T^{2} \)
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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

−11.14847417989891295739897483066, −10.53137836270269697952476272408, −10.33638119276043344783988095567, −8.010842027212615670849191779248, −6.78824030802854725280520930703, −6.11065537062175664955888256107, −5.15744314777416037199992522866, −4.27026764408609610712013095103, −2.54871545763813688467601505913, −0.40130407069677544474580825635, 2.02034555149708094118969903232, 4.19221022740119891425505947900, 4.98260536706607739820088715147, 5.87220179433164114194292144588, 6.62840939465465692948933370741, 7.73369378098228086069127693670, 9.218824942932474092958884733468, 10.76091432738478559805860792324, 11.15760122780300024224626477747, 12.10194256943937331286045791204

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