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Properties

Label 2-280-8.5-c1-0-19
Degree $2$
Conductor $280$
Sign $0.270 + 0.962i$
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  + (1.40 − 0.128i)2-s − 2.83i·3-s + (1.96 − 0.362i)4-s + i·5-s + (−0.364 − 3.99i)6-s + 7-s + (2.72 − 0.763i)8-s − 5.03·9-s + (0.128 + 1.40i)10-s + 4.54i·11-s + (−1.02 − 5.57i)12-s − 4.44i·13-s + (1.40 − 0.128i)14-s + 2.83·15-s + (3.73 − 1.42i)16-s − 8.18·17-s + ⋯
L(s)  = 1  + (0.995 − 0.0910i)2-s − 1.63i·3-s + (0.983 − 0.181i)4-s + 0.447i·5-s + (−0.148 − 1.62i)6-s + 0.377·7-s + (0.962 − 0.270i)8-s − 1.67·9-s + (0.0407 + 0.445i)10-s + 1.37i·11-s + (−0.296 − 1.60i)12-s − 1.23i·13-s + (0.376 − 0.0344i)14-s + 0.731·15-s + (0.934 − 0.356i)16-s − 1.98·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.270 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.270 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.270 + 0.962i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.270 + 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81942 - 1.37929i\)
\(L(\frac12)\) \(\approx\) \(1.81942 - 1.37929i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 + 0.128i)T \)
5 \( 1 - iT \)
7 \( 1 - T \)
good3 \( 1 + 2.83iT - 3T^{2} \)
11 \( 1 - 4.54iT - 11T^{2} \)
13 \( 1 + 4.44iT - 13T^{2} \)
17 \( 1 + 8.18T + 17T^{2} \)
19 \( 1 - 5.76iT - 19T^{2} \)
23 \( 1 + 0.380T + 23T^{2} \)
29 \( 1 - 0.968iT - 29T^{2} \)
31 \( 1 - 7.25T + 31T^{2} \)
37 \( 1 - 3.61iT - 37T^{2} \)
41 \( 1 + 1.35T + 41T^{2} \)
43 \( 1 + 1.16iT - 43T^{2} \)
47 \( 1 - 9.57T + 47T^{2} \)
53 \( 1 - 7.22iT - 53T^{2} \)
59 \( 1 + 5.89iT - 59T^{2} \)
61 \( 1 + 3.58iT - 61T^{2} \)
67 \( 1 + 9.25iT - 67T^{2} \)
71 \( 1 + 9.16T + 71T^{2} \)
73 \( 1 + 7.16T + 73T^{2} \)
79 \( 1 + 3.96T + 79T^{2} \)
83 \( 1 - 3.45iT - 83T^{2} \)
89 \( 1 + 14.6T + 89T^{2} \)
97 \( 1 + 1.74T + 97T^{2} \)
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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

−12.11012329451045745832795445046, −11.07445489407691483619432733725, −10.13929930131399738249182065876, −8.272967369756627541049451086676, −7.44108470923536558692922109899, −6.71550264965676296350025236652, −5.83175518279827465816202336237, −4.45489419204690496173046637968, −2.71095306367309679627254837544, −1.73134051647501358995959833334, 2.66186934162779188021598705818, 4.15492055623219584741704610882, 4.55955644586085193391624203746, 5.67218894689056437635857294472, 6.80090087548095191583715770420, 8.570568289456080013656853778173, 9.086145277001580176496403592548, 10.47778852736790357126843614601, 11.30030210768135393373857726937, 11.63675427046222894171076580244

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