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Properties

Label 2-280-280.67-c1-0-23
Degree $2$
Conductor $280$
Sign $0.954 - 0.297i$
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  + (−1.41 − 0.0333i)2-s + (0.388 + 1.44i)3-s + (1.99 + 0.0943i)4-s + (2.17 − 0.524i)5-s + (−0.500 − 2.06i)6-s + (2.60 − 0.445i)7-s + (−2.82 − 0.200i)8-s + (0.649 − 0.374i)9-s + (−3.09 + 0.669i)10-s + (1.03 − 1.79i)11-s + (0.638 + 2.93i)12-s + (−3.78 − 3.78i)13-s + (−3.70 + 0.542i)14-s + (1.60 + 2.94i)15-s + (3.98 + 0.376i)16-s + (−1.65 + 0.443i)17-s + ⋯
L(s)  = 1  + (−0.999 − 0.0235i)2-s + (0.224 + 0.836i)3-s + (0.998 + 0.0471i)4-s + (0.972 − 0.234i)5-s + (−0.204 − 0.841i)6-s + (0.985 − 0.168i)7-s + (−0.997 − 0.0707i)8-s + (0.216 − 0.124i)9-s + (−0.977 + 0.211i)10-s + (0.312 − 0.541i)11-s + (0.184 + 0.846i)12-s + (−1.04 − 1.04i)13-s + (−0.989 + 0.145i)14-s + (0.414 + 0.760i)15-s + (0.995 + 0.0942i)16-s + (−0.401 + 0.107i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.297i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 - 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.954 - 0.297i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.954 - 0.297i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13940 + 0.173290i\)
\(L(\frac12)\) \(\approx\) \(1.13940 + 0.173290i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0333i)T \)
5 \( 1 + (-2.17 + 0.524i)T \)
7 \( 1 + (-2.60 + 0.445i)T \)
good3 \( 1 + (-0.388 - 1.44i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (-1.03 + 1.79i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.78 + 3.78i)T + 13iT^{2} \)
17 \( 1 + (1.65 - 0.443i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.96 + 1.13i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.79 - 6.69i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 8.01T + 29T^{2} \)
31 \( 1 + (-3.76 - 2.17i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.81 - 1.02i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 1.76T + 41T^{2} \)
43 \( 1 + (7.50 - 7.50i)T - 43iT^{2} \)
47 \( 1 + (-1.06 - 0.286i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (8.24 - 2.20i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (8.33 + 4.81i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.39 - 4.27i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.23 - 0.866i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 5.34iT - 71T^{2} \)
73 \( 1 + (0.151 + 0.565i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-3.44 - 5.95i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.12 + 9.12i)T - 83iT^{2} \)
89 \( 1 + (6.29 - 3.63i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (10.7 + 10.7i)T + 97iT^{2} \)
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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

−11.51960039665873832585569091865, −10.69420588277250536516236226907, −9.796069462870330226932909752583, −9.345211873482698395741443774145, −8.251374082687502026509092637257, −7.29708207520061519399306740071, −5.88231193424634268052188982122, −4.83938833197474324762846920454, −3.12539436324073093704347767372, −1.52372459916353304291544574948, 1.73196283091660060405292882077, 2.29714910189924962735637847592, 4.77984392536438812064038430776, 6.27796398260527423006836488582, 7.10162433457706777652223851940, 7.86277220286983506240396432909, 9.004609204286809815788550560892, 9.782594865157039057148196283811, 10.72149643485242592294876978031, 11.80559410446672859377342292961

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