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Properties

Label 2-280-280.59-c1-0-7
Degree $2$
Conductor $280$
Sign $-0.0784 - 0.996i$
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  + (−0.881 − 1.10i)2-s + (−1.13 + 1.96i)3-s + (−0.446 + 1.94i)4-s + (1.70 + 1.45i)5-s + (3.17 − 0.477i)6-s + (−0.551 − 2.58i)7-s + (2.54 − 1.22i)8-s + (−1.07 − 1.86i)9-s + (0.105 − 3.16i)10-s + (−0.454 + 0.787i)11-s + (−3.32 − 3.09i)12-s + 5.66i·13-s + (−2.37 + 2.89i)14-s + (−4.78 + 1.69i)15-s + (−3.60 − 1.73i)16-s + (−3.14 + 5.45i)17-s + ⋯
L(s)  = 1  + (−0.623 − 0.781i)2-s + (−0.655 + 1.13i)3-s + (−0.223 + 0.974i)4-s + (0.760 + 0.649i)5-s + (1.29 − 0.195i)6-s + (−0.208 − 0.977i)7-s + (0.901 − 0.433i)8-s + (−0.359 − 0.622i)9-s + (0.0334 − 0.999i)10-s + (−0.137 + 0.237i)11-s + (−0.960 − 0.892i)12-s + 1.57i·13-s + (−0.634 + 0.772i)14-s + (−1.23 + 0.438i)15-s + (−0.900 − 0.434i)16-s + (−0.763 + 1.32i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0784 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0784 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.0784 - 0.996i$
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: \((2,\ 280,\ (\ :1/2),\ -0.0784 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.455205 + 0.492436i\)
\(L(\frac12)\) \(\approx\) \(0.455205 + 0.492436i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.881 + 1.10i)T \)
5 \( 1 + (-1.70 - 1.45i)T \)
7 \( 1 + (0.551 + 2.58i)T \)
good3 \( 1 + (1.13 - 1.96i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (0.454 - 0.787i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.66iT - 13T^{2} \)
17 \( 1 + (3.14 - 5.45i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.966 - 0.557i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.14 + 1.98i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.21iT - 29T^{2} \)
31 \( 1 + (1.41 - 2.45i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.789 - 1.36i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.163iT - 41T^{2} \)
43 \( 1 - 8.29iT - 43T^{2} \)
47 \( 1 + (-8.56 + 4.94i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.56 + 4.44i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-9.97 - 5.76i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.977 - 1.69i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.09 - 3.51i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.811iT - 71T^{2} \)
73 \( 1 + (-4.54 + 7.86i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (9.82 - 5.67i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.80T + 83T^{2} \)
89 \( 1 + (-12.4 + 7.16i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 9.99T + 97T^{2} \)
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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

−11.56951324013636342917785058696, −10.94408755220616181884916188541, −10.19387014651476106358915917938, −9.782657983578018517572604503840, −8.697701587009278266755513392413, −7.18349587468735277351703997968, −6.21836408946566832425373311831, −4.49304515041496188308029375138, −3.84454825231656119618189518689, −2.03956510622363711555750045842, 0.66991287080309818980257559082, 2.27541810391901633308840184171, 5.23906943608932422627869143233, 5.66681526499709525840338196776, 6.62783928993122860419926072167, 7.63205253582419278958020351179, 8.685355025281821331110573972597, 9.429026906524232691145153290402, 10.57180348038548870605754466850, 11.69369555925463281965093591889

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