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Properties

Label 2-280-56.19-c1-0-15
Degree $2$
Conductor $280$
Sign $0.980 + 0.197i$
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  + (−0.366 + 1.36i)2-s + (−1.5 + 0.866i)3-s + (−1.73 − i)4-s + (0.5 − 0.866i)5-s + (−0.633 − 2.36i)6-s + (0.866 − 2.5i)7-s + (2 − 1.99i)8-s + (0.999 + i)10-s + (−2.73 − 4.73i)11-s + 3.46·12-s + 1.26·13-s + (3.09 + 2.09i)14-s + 1.73i·15-s + (1.99 + 3.46i)16-s + (4.09 − 2.36i)17-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s + (−0.866 + 0.499i)3-s + (−0.866 − 0.5i)4-s + (0.223 − 0.387i)5-s + (−0.258 − 0.965i)6-s + (0.327 − 0.944i)7-s + (0.707 − 0.707i)8-s + (0.316 + 0.316i)10-s + (−0.823 − 1.42i)11-s + 0.999·12-s + 0.351·13-s + (0.827 + 0.560i)14-s + 0.447i·15-s + (0.499 + 0.866i)16-s + (0.993 − 0.573i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.197i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 + 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.980 + 0.197i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.980 + 0.197i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.686709 - 0.0683543i\)
\(L(\frac12)\) \(\approx\) \(0.686709 - 0.0683543i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.366 - 1.36i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.866 + 2.5i)T \)
good3 \( 1 + (1.5 - 0.866i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (2.73 + 4.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.26T + 13T^{2} \)
17 \( 1 + (-4.09 + 2.36i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.90 + 1.09i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.86 - 3.96i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.53iT - 29T^{2} \)
31 \( 1 + (4.09 + 7.09i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.73 + i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.19iT - 41T^{2} \)
43 \( 1 + 9.92T + 43T^{2} \)
47 \( 1 + (-4.73 + 8.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-8.83 + 5.09i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.09 - 0.633i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.86 + 6.69i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.69 - 8.13i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.73iT - 71T^{2} \)
73 \( 1 + (4.90 - 2.83i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.09 - 0.633i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.66iT - 83T^{2} \)
89 \( 1 + (-2.30 - 1.33i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 3.46iT - 97T^{2} \)
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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

−11.47796389514922498241097136580, −10.75711027395528413414110435465, −10.03236484815267748457529160494, −8.815518432857299043899645557484, −7.936188081983036929847304088329, −6.88099272370861388992834242266, −5.40807451643272543316684187045, −5.31148019118832097518367545871, −3.77049524703701860951555729538, −0.67892632169772755405947824905, 1.65275744805332902980747983456, 2.98656674267563331142827422427, 4.78262969922921196161621880485, 5.70470497488294204251490020701, 7.00230796136621082639941624582, 8.167508113869454100921636800297, 9.216452260799994384622982388506, 10.30726341680154756032416077254, 10.95230307640751841287099686411, 12.02485663851783246829251251338

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