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Properties

Label 2-280-56.19-c1-0-12
Degree $2$
Conductor $280$
Sign $0.553 + 0.832i$
Analytic cond. $2.23581$
Root an. cond. $1.49526$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·2-s + (−2.72 + 1.57i)3-s − 2.00·4-s + (−0.5 + 0.866i)5-s + (−2.22 − 3.85i)6-s + (−2.5 − 0.866i)7-s − 2.82i·8-s + (3.44 − 5.97i)9-s + (−1.22 − 0.707i)10-s + (2.44 + 4.24i)11-s + (5.44 − 3.14i)12-s + 0.449·13-s + (1.22 − 3.53i)14-s − 3.14i·15-s + 4.00·16-s + (−1.77 + 1.02i)17-s + ⋯
L(s)  = 1  + 0.999i·2-s + (−1.57 + 0.908i)3-s − 1.00·4-s + (−0.223 + 0.387i)5-s + (−0.908 − 1.57i)6-s + (−0.944 − 0.327i)7-s − 1.00i·8-s + (1.14 − 1.99i)9-s + (−0.387 − 0.223i)10-s + (0.738 + 1.27i)11-s + (1.57 − 0.908i)12-s + 0.124·13-s + (0.327 − 0.944i)14-s − 0.812i·15-s + 1.00·16-s + (−0.430 + 0.248i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.553 + 0.832i$
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: \(1\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.553 + 0.832i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (2.5 + 0.866i)T \)
good3 \( 1 + (2.72 - 1.57i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-2.44 - 4.24i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.449T + 13T^{2} \)
17 \( 1 + (1.77 - 1.02i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.67 + 2.12i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.949 + 0.548i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 10.0iT - 29T^{2} \)
31 \( 1 + (3.22 + 5.58i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3 + 1.73i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.36iT - 41T^{2} \)
43 \( 1 + 4.55T + 43T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.22 - 2.43i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (9.12 - 5.26i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.17 - 12.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.17 - 2.03i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.41iT - 71T^{2} \)
73 \( 1 + (2.32 - 1.34i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.674 - 0.389i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 12.2iT - 83T^{2} \)
89 \( 1 + (0.398 + 0.230i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.02iT - 97T^{2} \)
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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

−11.72202730138941023870165872258, −10.53899096652450263562332062058, −9.880706078934278131262986246366, −9.125906448384533328275340505616, −7.34801701601554728053257820494, −6.50948641537538032316247065784, −5.94408069937196695075850119713, −4.48135591222057677570545678753, −4.00471873064514068208803733396, 0, 1.48247515813452362094316225641, 3.45844640485722810805123464210, 4.97530387522793433700803239173, 5.94785832514809899906644857258, 6.79105016565175011509637243879, 8.340101778864542092392593481380, 9.281772559600105749965630978990, 10.68078958275007542786222917939, 11.09874401485015249161189485020

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