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Properties

Label 2-280-35.33-c1-0-1
Degree $2$
Conductor $280$
Sign $0.214 - 0.976i$
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  + (−2.75 − 0.739i)3-s + (−1.04 − 1.97i)5-s + (−1.91 + 1.82i)7-s + (4.46 + 2.57i)9-s + (2.37 + 4.11i)11-s + (1.92 − 1.92i)13-s + (1.42 + 6.22i)15-s + (−1.76 + 6.59i)17-s + (0.0439 − 0.0761i)19-s + (6.62 − 3.63i)21-s + (−0.242 + 0.0650i)23-s + (−2.81 + 4.13i)25-s + (−4.35 − 4.35i)27-s − 0.284i·29-s + (−3.69 + 2.13i)31-s + ⋯
L(s)  = 1  + (−1.59 − 0.426i)3-s + (−0.467 − 0.883i)5-s + (−0.722 + 0.691i)7-s + (1.48 + 0.859i)9-s + (0.715 + 1.23i)11-s + (0.534 − 0.534i)13-s + (0.368 + 1.60i)15-s + (−0.428 + 1.59i)17-s + (0.0100 − 0.0174i)19-s + (1.44 − 0.792i)21-s + (−0.0505 + 0.0135i)23-s + (−0.562 + 0.827i)25-s + (−0.838 − 0.838i)27-s − 0.0529i·29-s + (−0.662 + 0.382i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.214 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.214 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.214 - 0.976i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.214 - 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.333379 + 0.268215i\)
\(L(\frac12)\) \(\approx\) \(0.333379 + 0.268215i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.04 + 1.97i)T \)
7 \( 1 + (1.91 - 1.82i)T \)
good3 \( 1 + (2.75 + 0.739i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (-2.37 - 4.11i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.92 + 1.92i)T - 13iT^{2} \)
17 \( 1 + (1.76 - 6.59i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.0439 + 0.0761i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.242 - 0.0650i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 0.284iT - 29T^{2} \)
31 \( 1 + (3.69 - 2.13i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.05 - 3.93i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 9.16iT - 41T^{2} \)
43 \( 1 + (6.72 + 6.72i)T + 43iT^{2} \)
47 \( 1 + (-6.75 + 1.80i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.27 - 8.50i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.56 + 2.71i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.84 + 2.79i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.19 + 0.588i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 3.53T + 71T^{2} \)
73 \( 1 + (-8.72 - 2.33i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-10.7 - 6.18i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.1 - 10.1i)T - 83iT^{2} \)
89 \( 1 + (7.02 - 12.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.72 + 5.72i)T + 97iT^{2} \)
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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

−12.33507624068807015972424999616, −11.34329710251284263725582309365, −10.35899910047232676218546607568, −9.284946902387711618643413291324, −8.162639249255549720209348686030, −6.84628974124352108638162435330, −6.07738224248939864926479828275, −5.13070708902038722366333963995, −3.99675244932659134144574171803, −1.50121550094114329303911303560, 0.41897355543565401622353932707, 3.38495259605835855033251343408, 4.36514556219920557499482087894, 5.82095850087539355420435198582, 6.57253191133052149801014193240, 7.29811752402822374563855182502, 9.057941564791773326461020152363, 10.07117049739563344447469480106, 11.09254000192476806559970531934, 11.28226234843826576654818210655

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