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Properties

Label 2-280-35.17-c1-0-1
Degree $2$
Conductor $280$
Sign $-0.0701 - 0.997i$
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  + (−2.00 + 0.536i)3-s + (1.38 + 1.75i)5-s + (1.24 − 2.33i)7-s + (1.11 − 0.644i)9-s + (−3.09 + 5.36i)11-s + (0.782 + 0.782i)13-s + (−3.70 − 2.77i)15-s + (1.18 + 4.40i)17-s + (2.37 + 4.11i)19-s + (−1.25 + 5.33i)21-s + (−6.52 − 1.74i)23-s + (−1.17 + 4.86i)25-s + (2.50 − 2.50i)27-s + 5.30i·29-s + (2.16 + 1.25i)31-s + ⋯
L(s)  = 1  + (−1.15 + 0.309i)3-s + (0.618 + 0.785i)5-s + (0.472 − 0.881i)7-s + (0.372 − 0.214i)9-s + (−0.933 + 1.61i)11-s + (0.217 + 0.217i)13-s + (−0.957 − 0.715i)15-s + (0.286 + 1.06i)17-s + (0.544 + 0.943i)19-s + (−0.272 + 1.16i)21-s + (−1.35 − 0.364i)23-s + (−0.234 + 0.972i)25-s + (0.482 − 0.482i)27-s + 0.985i·29-s + (0.389 + 0.224i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0701 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0701 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.0701 - 0.997i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.0701 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.581850 + 0.624208i\)
\(L(\frac12)\) \(\approx\) \(0.581850 + 0.624208i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.38 - 1.75i)T \)
7 \( 1 + (-1.24 + 2.33i)T \)
good3 \( 1 + (2.00 - 0.536i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (3.09 - 5.36i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.782 - 0.782i)T + 13iT^{2} \)
17 \( 1 + (-1.18 - 4.40i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-2.37 - 4.11i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.52 + 1.74i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 5.30iT - 29T^{2} \)
31 \( 1 + (-2.16 - 1.25i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.77 + 6.61i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 2.51iT - 41T^{2} \)
43 \( 1 + (-0.404 + 0.404i)T - 43iT^{2} \)
47 \( 1 + (-8.63 - 2.31i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.66 + 9.92i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.0710 + 0.123i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.67 + 1.54i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.34 - 1.16i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 + (1.80 - 0.483i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-5.42 + 3.13i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.82 - 5.82i)T + 83iT^{2} \)
89 \( 1 + (4.58 + 7.94i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.64 + 2.64i)T - 97iT^{2} \)
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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

−12.03306654587233881553669177981, −10.85377676349941526577623801962, −10.39622275381182514390183155730, −9.830674785607603114039475071532, −7.997556823213468055768335000682, −7.09443868682102428494794789193, −6.03674763342431672084672767504, −5.11617053967063694881712379680, −3.95593727719972161846704722427, −1.94924954013853485646153355364, 0.75709618382549217796573967836, 2.70762310067522554414697897003, 4.89251794433030683467466333506, 5.65165373798636164484616399588, 6.13890615747949987851766214429, 7.84318830504581672968257819103, 8.704562352156873421152906326059, 9.742780136694783643868595023926, 10.96947540660644941754560688747, 11.67268019197092219557283884434

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