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Properties

Label 2-280-280.117-c1-0-34
Degree $2$
Conductor $280$
Sign $-0.683 + 0.730i$
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 + (0.5 + 0.133i)3-s + (1.73 + i)4-s + (0.133 − 2.23i)5-s + (−0.633 − 0.366i)6-s + (−2.5 − 0.866i)7-s + (−1.99 − 2i)8-s + (−2.36 − 1.36i)9-s + (−1 + 2.99i)10-s + (−2.36 + 1.36i)11-s + (0.732 + 0.732i)12-s + (3 − 3i)13-s + (3.09 + 2.09i)14-s + (0.366 − 1.09i)15-s + (1.99 + 3.46i)16-s + (−4.09 − 1.09i)17-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (0.288 + 0.0773i)3-s + (0.866 + 0.5i)4-s + (0.0599 − 0.998i)5-s + (−0.258 − 0.149i)6-s + (−0.944 − 0.327i)7-s + (−0.707 − 0.707i)8-s + (−0.788 − 0.455i)9-s + (−0.316 + 0.948i)10-s + (−0.713 + 0.411i)11-s + (0.211 + 0.211i)12-s + (0.832 − 0.832i)13-s + (0.827 + 0.560i)14-s + (0.0945 − 0.283i)15-s + (0.499 + 0.866i)16-s + (−0.993 − 0.266i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.683 + 0.730i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.683 + 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.683 + 0.730i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.683 + 0.730i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.224691 - 0.517821i\)
\(L(\frac12)\) \(\approx\) \(0.224691 - 0.517821i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.133 + 2.23i)T \)
7 \( 1 + (2.5 + 0.866i)T \)
good3 \( 1 + (-0.5 - 0.133i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (2.36 - 1.36i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3 + 3i)T - 13iT^{2} \)
17 \( 1 + (4.09 + 1.09i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3 - 1.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.5 + 5.59i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 2.46T + 29T^{2} \)
31 \( 1 + (0.169 - 0.0980i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.464 + 1.73i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 6.46iT - 41T^{2} \)
43 \( 1 + (-0.633 - 0.633i)T + 43iT^{2} \)
47 \( 1 + (-2.36 - 8.83i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3 + 11.1i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-4.26 + 2.46i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.96 + 3.40i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.96 - 2.13i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 8.53T + 71T^{2} \)
73 \( 1 + (2.90 - 10.8i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (0.803 + 0.464i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.56 - 7.56i)T - 83iT^{2} \)
89 \( 1 + (3.40 - 5.89i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.73 + 9.73i)T - 97iT^{2} \)
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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

−11.37745676860650656306730735772, −10.33571264910390650515867253691, −9.548068262084557867972703922450, −8.680675505428419110430477229338, −8.038453650913739986402008658978, −6.71258911076034949177717371278, −5.62129288273256550995335259040, −3.83836239787144667014960749441, −2.58328784326534667116533364894, −0.52592178290921678268321295084, 2.31307705145534951177027756132, 3.30170805076759981565462482078, 5.65363879878484313200229687128, 6.44834707022700743101092245593, 7.39156987478121972493192700625, 8.477334489636369314409119513331, 9.261129969932074242263183582365, 10.25557417694325049866736923695, 11.13244532522873837273466100496, 11.72621725502700865331489215070

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