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Properties

Label 2-280-280.59-c1-0-11
Degree $2$
Conductor $280$
Sign $0.138 - 0.990i$
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  + (−1.39 − 0.230i)2-s + (−0.929 + 1.60i)3-s + (1.89 + 0.642i)4-s + (2.18 + 0.457i)5-s + (1.66 − 2.03i)6-s + (1.10 + 2.40i)7-s + (−2.49 − 1.33i)8-s + (−0.226 − 0.392i)9-s + (−2.94 − 1.14i)10-s + (1.61 − 2.78i)11-s + (−2.79 + 2.45i)12-s − 0.668i·13-s + (−0.989 − 3.60i)14-s + (−2.77 + 3.09i)15-s + (3.17 + 2.43i)16-s + (1.62 − 2.81i)17-s + ⋯
L(s)  = 1  + (−0.986 − 0.162i)2-s + (−0.536 + 0.929i)3-s + (0.946 + 0.321i)4-s + (0.978 + 0.204i)5-s + (0.680 − 0.829i)6-s + (0.418 + 0.908i)7-s + (−0.881 − 0.471i)8-s + (−0.0755 − 0.130i)9-s + (−0.932 − 0.361i)10-s + (0.485 − 0.841i)11-s + (−0.806 + 0.707i)12-s − 0.185i·13-s + (−0.264 − 0.964i)14-s + (−0.715 + 0.799i)15-s + (0.793 + 0.608i)16-s + (0.393 − 0.681i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.138 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.138 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.138 - 0.990i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.138 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.650442 + 0.565885i\)
\(L(\frac12)\) \(\approx\) \(0.650442 + 0.565885i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 + 0.230i)T \)
5 \( 1 + (-2.18 - 0.457i)T \)
7 \( 1 + (-1.10 - 2.40i)T \)
good3 \( 1 + (0.929 - 1.60i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-1.61 + 2.78i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.668iT - 13T^{2} \)
17 \( 1 + (-1.62 + 2.81i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.77 - 3.33i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.41 - 7.64i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.17iT - 29T^{2} \)
31 \( 1 + (2.58 - 4.48i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.15 + 2.00i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 9.34iT - 41T^{2} \)
43 \( 1 + 6.84iT - 43T^{2} \)
47 \( 1 + (-3.12 + 1.80i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.0130 + 0.0225i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.0160 - 0.00925i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.897 - 1.55i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.14 + 2.39i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.13iT - 71T^{2} \)
73 \( 1 + (6.30 - 10.9i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-13.5 + 7.81i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.60T + 83T^{2} \)
89 \( 1 + (1.31 - 0.760i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.1T + 97T^{2} \)
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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

−11.66027208411296070418924990684, −10.85584317001534177858962409990, −10.27270994244341636254699009758, −9.212147884747128412938729705013, −8.737785097114788343020320103254, −7.25930393309060620568202422512, −5.92602049740697133987464512675, −5.31479766822787989034013089903, −3.39421570047353351594687728724, −1.84752280829492955116387295406, 1.04118334347645651100257501617, 2.15313003907106376160223005985, 4.61057036885500919340323643876, 6.24991515252308163816369438147, 6.64935720695290127403640013965, 7.64972677171583668201230378162, 8.770393635972966070086235120350, 9.785683918266457507153045152781, 10.61005117019823687295442961155, 11.46537875698653075714968843320

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