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Properties

Label 2-280-35.13-c1-0-4
Degree $2$
Conductor $280$
Sign $0.961 - 0.273i$
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  + (0.0703 − 0.0703i)3-s + (2.16 − 0.574i)5-s + (−0.562 + 2.58i)7-s + 2.99i·9-s + 0.777·11-s + (3.93 − 3.93i)13-s + (0.111 − 0.192i)15-s + (−0.982 − 0.982i)17-s + 1.14·19-s + (0.142 + 0.221i)21-s + (−1.46 − 1.46i)23-s + (4.34 − 2.48i)25-s + (0.421 + 0.421i)27-s + 4.69i·29-s + 6.45i·31-s + ⋯
L(s)  = 1  + (0.0405 − 0.0405i)3-s + (0.966 − 0.256i)5-s + (−0.212 + 0.977i)7-s + 0.996i·9-s + 0.234·11-s + (1.09 − 1.09i)13-s + (0.0288 − 0.0496i)15-s + (−0.238 − 0.238i)17-s + 0.263·19-s + (0.0310 + 0.0482i)21-s + (−0.304 − 0.304i)23-s + (0.868 − 0.496i)25-s + (0.0810 + 0.0810i)27-s + 0.870i·29-s + 1.15i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.273i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 - 0.273i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.961 - 0.273i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.961 - 0.273i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49043 + 0.207980i\)
\(L(\frac12)\) \(\approx\) \(1.49043 + 0.207980i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-2.16 + 0.574i)T \)
7 \( 1 + (0.562 - 2.58i)T \)
good3 \( 1 + (-0.0703 + 0.0703i)T - 3iT^{2} \)
11 \( 1 - 0.777T + 11T^{2} \)
13 \( 1 + (-3.93 + 3.93i)T - 13iT^{2} \)
17 \( 1 + (0.982 + 0.982i)T + 17iT^{2} \)
19 \( 1 - 1.14T + 19T^{2} \)
23 \( 1 + (1.46 + 1.46i)T + 23iT^{2} \)
29 \( 1 - 4.69iT - 29T^{2} \)
31 \( 1 - 6.45iT - 31T^{2} \)
37 \( 1 + (1.30 - 1.30i)T - 37iT^{2} \)
41 \( 1 + 9.81iT - 41T^{2} \)
43 \( 1 + (7.13 + 7.13i)T + 43iT^{2} \)
47 \( 1 + (7.34 + 7.34i)T + 47iT^{2} \)
53 \( 1 + (2.08 + 2.08i)T + 53iT^{2} \)
59 \( 1 + 8.29T + 59T^{2} \)
61 \( 1 - 5.88iT - 61T^{2} \)
67 \( 1 + (6.30 - 6.30i)T - 67iT^{2} \)
71 \( 1 - 12.3T + 71T^{2} \)
73 \( 1 + (7.23 - 7.23i)T - 73iT^{2} \)
79 \( 1 - 2.83iT - 79T^{2} \)
83 \( 1 + (-10.4 + 10.4i)T - 83iT^{2} \)
89 \( 1 + 3.41T + 89T^{2} \)
97 \( 1 + (8.50 + 8.50i)T + 97iT^{2} \)
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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

−12.01658153955834211099627289393, −10.77718301399364542497819864241, −10.11580298894858462886619327973, −8.867679062433316866698540752266, −8.377604536588980642069589090085, −6.85239777638289193354574670256, −5.70526341845050763276363944209, −5.08609449576980282883125284739, −3.15616482518118435619939941997, −1.82077648644162062331303394644, 1.47809747834376396348206724448, 3.35296548174023905408188347956, 4.43734747124272343605227394520, 6.27479766120075642186171870791, 6.47488128706716714946284290182, 7.934730608885520569248953607504, 9.342431979349845579707064902451, 9.689071959361460078243787855950, 10.90680354661103394567845099990, 11.64601074336031151805266042727

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