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Properties

Label 2-280-56.19-c1-0-9
Degree $2$
Conductor $280$
Sign $0.197 - 0.980i$
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  + (1.36 + 0.366i)2-s + (−1.5 + 0.866i)3-s + (1.73 + i)4-s + (0.5 − 0.866i)5-s + (−2.36 + 0.633i)6-s + (−0.866 + 2.5i)7-s + (1.99 + 2i)8-s + (1 − 0.999i)10-s + (0.732 + 1.26i)11-s − 3.46·12-s + 4.73·13-s + (−2.09 + 3.09i)14-s + 1.73i·15-s + (1.99 + 3.46i)16-s + (−1.09 + 0.633i)17-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + (−0.866 + 0.499i)3-s + (0.866 + 0.5i)4-s + (0.223 − 0.387i)5-s + (−0.965 + 0.258i)6-s + (−0.327 + 0.944i)7-s + (0.707 + 0.707i)8-s + (0.316 − 0.316i)10-s + (0.220 + 0.382i)11-s − 0.999·12-s + 1.31·13-s + (−0.560 + 0.827i)14-s + 0.447i·15-s + (0.499 + 0.866i)16-s + (−0.266 + 0.153i)17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.197 - 0.980i$
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: \((2,\ 280,\ (\ :1/2),\ 0.197 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35979 + 1.11359i\)
\(L(\frac12)\) \(\approx\) \(1.35979 + 1.11359i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 - 0.366i)T \)
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 \( 1 + (-0.732 - 1.26i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.73T + 13T^{2} \)
17 \( 1 + (1.09 - 0.633i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (7.09 + 4.09i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.13 - 2.96i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 10.4iT - 29T^{2} \)
31 \( 1 + (-1.09 - 1.90i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.73 - i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.19iT - 41T^{2} \)
43 \( 1 - 3.92T + 43T^{2} \)
47 \( 1 + (-1.26 + 2.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.169 + 0.0980i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.09 + 2.36i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.13 + 3.69i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.69 + 9.86i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.26iT - 71T^{2} \)
73 \( 1 + (10.0 - 5.83i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.09 + 2.36i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 14.6iT - 83T^{2} \)
89 \( 1 + (-12.6 - 7.33i)T + (44.5 + 77.0i)T^{2} \)
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

−12.05323045596012741862581138491, −11.28821988870000966367566783489, −10.58749562555485241649536940222, −9.164341836463628037801153624568, −8.224070132457629828057656207654, −6.56792973664700916856743705831, −5.95454125217632538908522248559, −5.02006561977293395249861964572, −4.04882028821424225776011425544, −2.35168056102284378835320241292, 1.25241552587241277003225358824, 3.22754178116673726826814580947, 4.33069378762928077674375580416, 5.84353752345970497611552837095, 6.43450984086770305383163460526, 7.16189337562416938943674059295, 8.791728144548701526497460815535, 10.44458208811002636212049187168, 10.84310148461305090376729516153, 11.62114788921739078934851553240

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