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Properties

Label 2-280-280.107-c1-0-31
Degree $2$
Conductor $280$
Sign $0.261 + 0.965i$
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  + (−0.0697 + 1.41i)2-s + (−0.767 − 0.205i)3-s + (−1.99 − 0.197i)4-s + (0.436 − 2.19i)5-s + (0.343 − 1.06i)6-s + (−1.34 + 2.28i)7-s + (0.417 − 2.79i)8-s + (−2.05 − 1.18i)9-s + (3.06 + 0.770i)10-s + (−2.69 − 4.66i)11-s + (1.48 + 0.560i)12-s + (0.119 + 0.119i)13-s + (−3.12 − 2.05i)14-s + (−0.786 + 1.59i)15-s + (3.92 + 0.784i)16-s + (1.19 − 4.46i)17-s + ⋯
L(s)  = 1  + (−0.0493 + 0.998i)2-s + (−0.443 − 0.118i)3-s + (−0.995 − 0.0985i)4-s + (0.195 − 0.980i)5-s + (0.140 − 0.436i)6-s + (−0.506 + 0.862i)7-s + (0.147 − 0.989i)8-s + (−0.683 − 0.394i)9-s + (0.969 + 0.243i)10-s + (−0.811 − 1.40i)11-s + (0.429 + 0.161i)12-s + (0.0330 + 0.0330i)13-s + (−0.835 − 0.548i)14-s + (−0.202 + 0.411i)15-s + (0.980 + 0.196i)16-s + (0.290 − 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.261 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.261 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.261 + 0.965i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.261 + 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.403740 - 0.308982i\)
\(L(\frac12)\) \(\approx\) \(0.403740 - 0.308982i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0697 - 1.41i)T \)
5 \( 1 + (-0.436 + 2.19i)T \)
7 \( 1 + (1.34 - 2.28i)T \)
good3 \( 1 + (0.767 + 0.205i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (2.69 + 4.66i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.119 - 0.119i)T + 13iT^{2} \)
17 \( 1 + (-1.19 + 4.46i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.01 + 0.584i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.442 - 0.118i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 5.92T + 29T^{2} \)
31 \( 1 + (6.46 - 3.73i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.20 + 4.50i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 7.53T + 41T^{2} \)
43 \( 1 + (-3.70 + 3.70i)T - 43iT^{2} \)
47 \( 1 + (-2.68 - 10.0i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.0105 - 0.0393i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (3.51 - 2.02i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.68 + 4.43i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.494 + 1.84i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 9.18iT - 71T^{2} \)
73 \( 1 + (-4.84 - 1.29i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-1.88 + 3.25i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.95 + 2.95i)T - 83iT^{2} \)
89 \( 1 + (-5.18 - 2.99i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-12.9 - 12.9i)T + 97iT^{2} \)
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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

−11.95564096750379234287938066865, −10.65506522184473155683665254272, −9.232464193316312918766551586022, −8.854712672698896332550187461172, −7.890890336617377595333698097315, −6.41601183398730606068556929326, −5.64397425485317207303777744790, −5.04141833803660708481803373189, −3.19783818588853318349847772746, −0.40036238190715887560200274489, 2.16634860632395520936526264936, 3.43836748261295172750957008659, 4.67733766816310372371538326321, 5.94246038390374943065099040044, 7.23182617973954240716862340059, 8.260665736911301529834494776666, 9.804236900316961763682521050490, 10.36103620340962754914048815036, 10.84757020167728610744349786862, 11.93548610037528992052869234630

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