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Properties

Label 2-280-7.2-c1-0-5
Degree $2$
Conductor $280$
Sign $0.991 + 0.126i$
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  + (0.5 + 0.866i)3-s + (0.5 − 0.866i)5-s + (0.5 − 2.59i)7-s + (1 − 1.73i)9-s + (1 + 1.73i)11-s + 4·13-s + 0.999·15-s + (−3 + 5.19i)19-s + (2.5 − 0.866i)21-s + (−1.5 + 2.59i)23-s + (−0.499 − 0.866i)25-s + 5·27-s − 3·29-s + (−0.999 + 1.73i)33-s + (−2 − 1.73i)35-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (0.223 − 0.387i)5-s + (0.188 − 0.981i)7-s + (0.333 − 0.577i)9-s + (0.301 + 0.522i)11-s + 1.10·13-s + 0.258·15-s + (−0.688 + 1.19i)19-s + (0.545 − 0.188i)21-s + (−0.312 + 0.541i)23-s + (−0.0999 − 0.173i)25-s + 0.962·27-s − 0.557·29-s + (−0.174 + 0.301i)33-s + (−0.338 − 0.292i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.991 + 0.126i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52754 - 0.0969384i\)
\(L(\frac12)\) \(\approx\) \(1.52754 - 0.0969384i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.5 + 2.59i)T \)
good3 \( 1 + (-0.5 - 0.866i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3 - 5.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6 + 10.3i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 7T + 41T^{2} \)
43 \( 1 + 9T + 43T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5 - 8.66i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.5 + 9.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10T + 71T^{2} \)
73 \( 1 + (-4 - 6.92i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3 - 5.19i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3T + 83T^{2} \)
89 \( 1 + (8.5 - 14.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−11.84209759969877538035208474160, −10.68744271651203120512119314324, −9.969614304350331885090936122221, −9.085795471971761816262852933449, −8.089326253090304074177819828932, −6.93908139014821479229079562924, −5.81835868962309026392036215833, −4.28594130121994344478204656725, −3.69355667433550680345315236611, −1.48216594520411328352741169624, 1.83232355697355770878513762232, 3.08037391997683798988042195968, 4.75173078951368002675729806368, 6.06103251015962913007064911157, 6.85581568554923371841397739720, 8.254349299597378995482278508418, 8.723048765687327553149368847023, 10.01621194868901967206026104277, 11.08944934396701605930796412373, 11.75937320688817184309430848931

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